# Local inverses and distance-weighted elliptic estimates

Ellipticity gives a local inverse with exactly as many derivatives as the order of the operator. The useful question here is how little regularity its coefficients may have. Continuity will suffice for the highest-order coefficients. Lower-order coefficients can be unbounded; their admissible integrability depends on how many derivatives separate their term from the principal part.

We first build a constant-coefficient solver and record its quantitative bounds. Multiplication estimates then determine the coefficient assumptions. A separate localization argument gives estimates on arbitrary open subsets of one fixed neighborhood, with constants independent of the subset. Finally, a commutator argument supplies the missing highest derivatives when the equation is initially defined only as a distribution.

## 1. Conventions and prerequisite contracts

Let \(n,m\geq1\) be integers and \(1<p<\infty\). The order \(m\) is arbitrary. Coefficients and functions may be complex-valued. We use the Fourier convention of Section 1 of [Fourier transforms, finite spectra and convex separation](prerequisite-bridges.md):

\[
D_j=-i\partial_j,\qquad
\widehat f(\xi)=\int e^{-ix\cdot\xi}f(x)\,dx,\qquad
\mathcal F^{-1}h(x)=(2\pi)^{-n}\int e^{ix\cdot\xi}h(\xi)\,d\xi.
\]

The space \(W^{s,p}(V)\), for a nonnegative integer \(s\), consists of distributions whose derivatives of orders at most \(s\) belong to \(L^p(V)\). We write \(\nabla^j u\) for the finite array \((D^\alpha u)_{|\alpha|=j}\), using its pointwise \(\ell^p\) norm inside an \(L^p\) norm. Switching to the sum of the component norms changes only constants depending on \(n,m,p\).

The proofs use the following entry results, each with its stated hypotheses.

* **Distribution operations:** Distributions carry the usual test-function topology. Multiplication by a smooth function and distributional differentiation are continuous operations. A distribution supported in a compact set has finite order: after inserting a fixed cutoff near its support, its pairing is bounded by a constant times finitely many suprema of derivatives of the test function on a fixed compact neighborhood. It therefore acts on smooth functions through that cutoff and is tempered. Convolution is defined when one factor is compactly supported; differentiation transfers between its factors, and associativity holds in the uses here where all but at most one factor are compactly supported. Convolution of a tempered distribution with a compactly supported distribution is tempered. A smooth kernel paired with a compactly supported distribution gives a smooth function, also locally when all relevant kernel arguments stay away from its singular set. Fourier inversion of a compactly supported distribution is smooth, with derivatives obtained by differentiating its exponential pairing. These distribution-calculus entry facts are used in (L4)–(L5), (L12), (L14), and the smoothness argument at the end of Section 2. Sections 13.6–13.10 of [Detecting regularity without choosing coordinates](geometric-microlocal-calculus.md) give the complete proofs of this interface, including the exact applications to the inverse identities below.
* **Integration and weak derivatives:** Lebesgue integration, Hölder and Young inequalities, Tonelli, dominated and monotone convergence; completeness of \(L^p\); density of \(C_c^\infty(\mathbb R^n)\) in \(L^p\) for finite \(p\); translation continuity and convergence of smooth approximate identities in these spaces. Locally, smooth approximation holds in integer \(W^{s,p}\). A distribution represented by an \(L^p\) limit has the corresponding distributional derivatives. These are the integration and weak-derivative facts used below. Sections 15.1–15.5 of [Banach estimates, quotient spaces and compact parameter arguments](banach-foundation-bridges.md) prove these operations with their full norms, scale factors and derivative signs, and identify their exact applications below.
* **Fourier multiplier estimate:** If \(b\in C^{n+2}(\mathbb R^n\setminus\{0\})\) and
  \[
  \max_{|\beta|\leq n+2}\sup_{\xi\ne0}
  |\xi|^{|\beta|}|\partial_\xi^\beta b(\xi)|\leq B,
  \]
  then \(b(D)\) extends from Schwartz functions to \(L^p(\mathbb R^n)\), with norm at most \(C_{n,p}B\). The source is Terence Tao, *Lecture Notes 4 for 247A*, Theorem 4.4, pp. 19–20; its singular-integral implication is Corollary 2.10, pp. 8–9. The bound uses only finitely many symbol derivatives. Section 13 proves this statement with the original operator, finite exponents and full constants; its Fourier maps are constructed in Section 7 of [Fourier transforms, finite spectra and convex separation](prerequisite-bridges.md).
* **Fractional-integration estimate:** For \(0<s<n\), \(1<p<q<\infty\), and \(1/q=1/p-s/n\), convolution with \(|x|^{s-n}\) maps \(L^p(\mathbb R^n)\) to \(L^q(\mathbb R^n)\). The source is Tao, *Lecture Notes 2 for 247A*, Corollary 6.3, p. 16, with Proposition 6.1 on p. 15. No strong \(L^\infty\) endpoint is included. Section 12 proves this original kernel estimate, including its finite endpoint, full constants and derivative-array application.
* **Lipschitz product rule:** A Lipschitz scalar function has weak first derivatives in \(L^\infty\), bounded by its Lipschitz constant, and satisfies the weak product rule with smooth functions. This local fact also permits integration by parts against compactly supported smooth functions. It will be used only in the weak-coefficient part. Section 11 supplies the complete weak-derivative, product, cutoff and commutator proofs for the original complex coefficients.

There is also an earlier **constant-strength antecedent**: Hörmander, *The Analysis of Linear Partial Differential Operators II*, Theorem 13.2.1. For an operator with nonzero frozen symbol, continuous coefficients and constant strength near a point, it provides a local \(L^2\) right inverse, a left inverse on compactly supported smooth functions, and bounded \(Q(D)E\) when the constant-coefficient operator \(Q\) is weaker than the frozen operator. In this terminology, write
\(\widetilde q(\xi)^2=\sum_\beta|\partial_\xi^\beta q(\xi)|^2\);
“weaker” means \(\widetilde q\leq C\widetilde p\), and constant strength means pairwise comparability of the frozen \(\widetilde p_x\). This antecedent has an explicit dependency on that volume's constant-coefficient inverse theorem 10.3.7. It does not supply the rough lower-coefficient \(L^p\) theorem below. We give the elliptic construction needed here directly, leaving the more general constant-strength contract separate.

The equation in that antecedent has a specific interpretation. The finite-dimensional weaker-operator representation is
\(P(x,D)=P_0(D)+\sum_{\nu=1}^r c_\nu(x)P_\nu(D)\), with \(P_0\) the frozen operator, \(P_\nu\) weaker than \(P_0\), and continuous \(c_\nu\) vanishing at the frozen point. Define \(PEf=P_0Ef+\sum_\nu c_\nu P_\nu Ef\). Each \(P_\nu Ef\), including \(P_0Ef\), belongs to \(L^2\) by the stated operator bounds; after shrinking the neighborhood, its product with \(c_\nu\) is an \(L^2\) function. This is the interpretation in the first remark following Theorem 13.2.1, not multiplication of arbitrary distributions by continuous coefficients.

The proofs below establish the fundamental solution, the local Hölder multiplier bound, the perturbation argument, the distance-weighted estimate, and the Lipschitz commutator. The Banach-space geometric-series argument is also available in Section 3 of [Finite defects under perturbation](fredholm-stability.md); we recall its short application where it is needed.

## 2. A frozen solver without a restriction on dimension or order

Let \(q(\xi)\) be homogeneous of degree \(m\), with

\[
|q(\xi)|\geq c|\xi|^m\quad(\xi\in\mathbb R^n).                 \tag{L1}
\]

Choose \(\chi\in C_c^\infty(\mathbb R^n)\), equal to one for \(|\xi|\leq1/2\) and zero for \(|\xi|\geq1\). Set

\[
b_0(\xi)=\frac{1-\chi(\xi)}{q(\xi)},\qquad F_0=\mathcal F^{-1}b_0.
                                                                    \tag{L2}
\]

The quotient is smooth across zero because its numerator vanishes there. Repeated differentiation of a reciprocal, or induction in the identity \(q/q=1\), gives

\[
|\partial^\beta b_0(\xi)|\leq C_\beta(1+|\xi|)^{-m-|\beta|}.
                                                                    \tag{L3}
\]

In particular \(F_0\) is tempered and
\(q(D)F_0=\delta+\omega\), where \(\omega=-\mathcal F^{-1}\chi\) is Schwartz.

Here is an exact fundamental solution. When \(m<n\), the locally integrable function \(1/q\) defines a tempered distribution \(T\). When \(m\geq n\), put \(L=m-n\) and define, for \(\varphi\in\mathcal S\),

\[
\langle T,\varphi\rangle=
\int_{\mathbb R^n}\frac{\displaystyle\varphi(\xi)-\chi(\xi)
  \sum_{|\beta|\leq L}\frac{\partial^\beta\varphi(0)}{\beta!}\xi^\beta}
 {q(\xi)}\,d\xi.                                                   \tag{L4}
\]

Taylor's theorem bounds the numerator by \(C|\xi|^{L+1}\) near zero. Its quotient is integrable there, since the radial power after including the volume element is \(n-1-m+L+1=0\). At infinity, Schwartz decay applies. These same estimates, in terms of finitely many Schwartz seminorms, prove that \(T\) is tempered. As \(L<m\), all derivatives of \(q\varphi\) of orders at most \(L\) vanish at zero. Consequently \(qT=1\). Thus

\[
G=\mathcal F^{-1}T,\qquad q(D)G=\delta.                            \tag{L5}
\]

Moreover, \(T-b_0\) is supported in \(\operatorname{supp}\chi\). Its inverse Fourier transform \(H=G-F_0\) is smooth: on a fixed compact frequency set one may differentiate the exponential in its distributional pairing any number of times. Those differentiated pairings depend continuously on \(x\), by the finite-order bound for a compactly supported distribution. This proves smoothness, without assuming that \(G\) is an ordinary homogeneous function. In the dimensions where logarithmic terms occur, (L4) includes them automatically.

We need two kernel estimates. Fix \(|\alpha|=j\leq m\) and put \(k=m-j\). The symbol of \(D^\alpha F_0\) is \(\xi^\alpha b_0(\xi)\). With
\(\psi(\xi)=\chi(\xi/2)-\chi(\xi)\), its decomposition into annuli has inverse transforms

\[
K_\ell(x)=2^{\ell(n-k)}\kappa_\alpha(2^\ell x),\quad
\kappa_\alpha=\mathcal F^{-1}\!\left(\psi(\xi)\frac{\xi^\alpha}{q(\xi)}\right),
\quad \ell=0,1,\ldots.                                             \tag{L6}
\]

The function \(\kappa_\alpha\) is Schwartz. Indeed its Fourier transform is smooth with compact support away from zero; multiplication by powers of \(x\) in the inverse transform is integration by parts. The telescoping identity \(\sum_{\ell\geq0}\psi(2^{-\ell}\xi)=1-\chi(\xi)\) proves (L6) in tempered distributions. For every integer \(N\),

\[
|K_\ell(x)|\leq C_N2^{\ell(n-k)}(1+2^\ell|x|)^{-N}.              \tag{L7}
\]

If \(k>0\), then \(\sum_\ell\|K_\ell\|_1\leq C\sum_\ell2^{-\ell k}<\infty\). It therefore represents an \(L^1\) kernel. Splitting the sum at \(2^\ell|x|=1\) gives, for \(0<|x|\leq1\),

\[
|D^\alpha F_0(x)|\leq C
\begin{cases}
|x|^{k-n},&0<k<n,\\
1+|\log|x||,&k=n,\\
1,&k>n.
\end{cases}                                                       \tag{L8}
\]

For example the terms below the splitting point form a geometric sum when \(k\ne n\); when \(k=n\), their number is at most \(C(1+|\log|x||)\). Terms above the splitting point form a convergent geometric tail if \(N>n-k\). When \(|x|\geq1\), (L7), with \(N\) as large as desired, gives rapid decay. These arguments also prove smoothness away from zero by differentiating (L7). In particular,

\[
D^\alpha F_0\in L^r_{\mathrm{loc}}
\quad\hbox{if}\quad k>n(1-1/r),\qquad 1\leq r\leq\infty.          \tag{L9}
\]

For \(r<\infty\) this follows by integrating \(\rho^{r(k-n)+n-1}\); the logarithmic case is integrable to every finite power. The stated condition for \(r=\infty\) is \(k>n\). No assertion that a highest derivative is an integrable kernel is being made. When \(k=0\), its boundedness on \(L^p\) follows from Fourier multiplier estimate and (L3).

For \(0<k<n/p\), (L8), its decay at infinity, and Fractional-integration estimate give the finite endpoint

\[
\|D^\alpha F_0*g\|_q\leq C\|g\|_p,
\qquad \frac1q=\frac1p-\frac{k}{n}.                              \tag{L10}
\]

Together with the multiplier case \(k=0,q=p\), this is the global endpoint estimate whenever the displayed \(q\) is finite. For strictly larger \(1/q\), Young's inequality applies to a localized kernel: choose \(r\) by \(1+1/q=1/p+1/r\) and use (L9). This includes \(q=\infty\) precisely when \(k>n/p\). Interpolation with, or the finite-measure inclusion from, the finite endpoint handles the other \(p\leq q<\infty\).

Consequently, if \(V\subset B_1\), extend \(g\in L^p(V)\) by zero and define
\(S_Vg=(G*g)|_V\). For every \(j\leq m\),

\[
\|D^\alpha S_Vg\|_{L^q(V)}\leq C\|g\|_{L^p(V)},\qquad
p\leq q\leq\infty,\quad
\frac1q\geq\frac1p-\frac{k}{n},                                  \tag{L11}
\]

where the last inequality must be strict if \(q=\infty\). The constant is independent of the particular \(V\subset B_1\), with the frozen polynomial and the chosen exponents fixed. To verify the smooth correction, all arguments \(x-y\), for \(x,y\in V\), lie in \(B_2\); hence each derivative of \(H\) is bounded there and its convolution is bounded by \(C\|g\|_1\leq C|B_1|^{1-1/p}\|g\|_p\). Its \(L^q(V)\) norm has the required uniform bound. The same reasoning localizes the kernels in the strict Young estimates.

Both inverse identities are exact:

\[
q(D)S_Vg=g\text{ in }V,\qquad S_Vq(D)v=v\quad(v\in C_c^\infty(V)).
                                                                    \tag{L12}
\]

For the first, convolve (L5) with the compactly supported distribution given by the zero extension of \(g\). For the second, differentiation may be transferred to the compactly supported smooth factor: \(G*q(D)v=(q(D)G)*v=v\).

For completeness, any other fundamental solution differs from \(G\) by a smooth function. If \(q(D)w=0\), take a cutoff \(\eta\) equal to one near any prescribed compact set. Then
\[
\eta w=F_0*q(D)(\eta w)-\omega*(\eta w).
\]
Here \(q(D)(\eta w)=[q(D),\eta]w\) has compact support away from that set. The first convolution is smooth near the set because \(F_0\) is smooth away from zero; the second is smooth because \(\omega\) is smooth and \(\eta w\) has compact support. This proves the assertion locally everywhere. It also explains the smooth-remainder assertion for a regularized inverse with a different low-frequency cutoff.

## 3. A parameter in the constant-coefficient estimate

**Lemma.** For a homogeneous elliptic \(q\) of degree \(m\),

\[
\sum_{|\alpha|\leq m} A^{m-|\alpha|}\|D^\alpha v\|_p
\leq C\bigl(\|q(D)v\|_p+A^m\|v\|_p\bigr),\qquad A>0,\quad
v\in W^{m,p}(\mathbb R^n).                                        \tag{L13}
\]

The constant is independent of \(A\). It can also be chosen uniformly when \(q\) ranges over a compact set of homogeneous elliptic polynomials of degree \(m\).

**Proof.** At \(A=1\), the Fourier identity \(q b_0=1+\widehat\omega\) gives
\[
D^\alpha v=(D^\alpha F_0)*q(D)v-(D^\alpha\omega)*v.                \tag{L14}
\]
It holds first on Schwartz functions. Each first operator is bounded on \(L^p\) by the multiplier contract for top order, and by the \(L^1\) kernel estimate for lower orders. The second kernel is Schwartz. Smooth cutoff and mollification approximate any \(W^{m,p}\) function in that norm, so the identity and estimate extend to the stated domain. For \(w(y)=v(y/A)\), the derivative norm is
\(\|D^\alpha w\|_p=A^{n/p-|\alpha|}\|D^\alpha v\|_p\),
while homogeneity gives
\(\|q(D)w\|_p=A^{n/p-m}\|q(D)v\|_p\).
Apply the estimate at one and multiply by \(A^{m-n/p}\).

Finally, a compact family of elliptic polynomials has a positive common minimum of \(|q(\theta)|\) on the unit sphere. Its coefficients are bounded. The reciprocal differentiation identity bounds every required derivative in (L3) uniformly; the integrations by parts and the multiplier theorem use only finitely many of these bounds. The same \(C\) therefore works for the family. \(\square\)

## 4. The coefficient table and a local inverse in \(L^p\)

Consider
\[
P=\sum_{|\alpha|\leq m}a_\alpha(x)D^\alpha,
\qquad P_m=\sum_{|\alpha|=m}a_\alpha(x)D^\alpha                     \tag{L15}
\]
near zero. Assume that the coefficients of \(P_m\) are continuous and its frozen symbol
\(q(\xi)=\sum_{|\alpha|=m}a_\alpha(0)\xi^\alpha\) is elliptic. For every lower-order term set \(k=m-|\alpha|>0\) and require the following local integrability.

| Derivative gap \(k\) | Coefficient space \(L^{r_\alpha}_{\rm loc}\) | Exponent used for \(D^\alpha S_Vg\) |
| --- | --- | --- |
| \(k<n/p\) | \(r_\alpha=n/k\) | \(1/q_\alpha=1/p-k/n\) |
| \(k=n/p\) | \(r_\alpha=p+\varepsilon_\alpha\), some \(\varepsilon_\alpha>0\) | \(q_\alpha=p(p+\varepsilon_\alpha)/\varepsilon_\alpha\) |
| \(k>n/p\) | \(r_\alpha=p\) | \(q_\alpha=\infty\) |

There are finitely many coefficients, so different positive \(\varepsilon_\alpha\) cause no difficulty. At the critical gap any finite \(q_\alpha\) is allowed by (L11). The table gives \(1/p=1/r_\alpha+1/q_\alpha\) in every row.

**Theorem.** Every sufficiently small open neighborhood \(V\) of zero has a linear operator \(E:L^p(V)\to L^p(V)\) satisfying

\[
PEf=f\quad(f\in L^p(V)),\qquad EPv=v\quad(v\in C_c^\infty(V)),     \tag{L16}
\]

and, simultaneously for all \(|\alpha|\leq m\),

\[
D^\alpha E:L^p(V)\longrightarrow L^q(V)\text{ bounded if }
p\leq q\leq\infty,\quad
\frac1q\geq\frac1p-\frac{m-|\alpha|}{n},                           \tag{L17}
\]

with strict inequality when \(q=\infty\). The equation in (L16) is a distributional equality and every term of \(PEf\) is represented by an \(L^p\) function. In particular \(Ef\in W^{m,p}(V)\).

**Proof.** Use \(S_V\) from (L11), with the same frozen polynomial, and compute
\[
PS_V=I+R_V,\qquad
R_Vg=\sum_{|\alpha|=m}(a_\alpha-a_\alpha(0))D^\alpha S_Vg
       +\sum_{|\alpha|<m}a_\alpha D^\alpha S_Vg.                   \tag{L18}
\]
The top-order terms have norms at most
\(C\sup_V|a_\alpha-a_\alpha(0)|\|g\|_p\).
For a lower-order term, Hölder's inequality and the corresponding row of the table give
\[
\|a_\alpha D^\alpha S_Vg\|_p
\leq C\|a_\alpha\|_{L^{r_\alpha}(V)}\|g\|_p.                      \tag{L19}
\]
All \(r_\alpha\) are finite. On shrinking the surrounding ball, the first coefficients tend to zero by continuity and the norms in (L19) tend to zero by absolute continuity of their integrals. The constants in (L11) do not grow as \(V\) shrinks inside \(B_1\). Choose the ball so that \(\|R_V\|\leq1/2\) for every \(V\) contained in it.

The series \(B_V=\sum_{\nu=0}^\infty(-R_V)^\nu\) converges in operator norm, since \(L^p(V)\) is Banach. Multiplication of its finite partial sums by \(I+R_V\) leaves an error of norm at most \(2^{-N}\), so \(B_V\) is its two-sided inverse and \(\|B_V\|\leq2\). Define
\[
E=S_VB_V.                                                        \tag{L20}
\]
Equations (L18) and (L11) give the right identity and every bound in (L17). They also justify all coefficient products used in the equation.

For the left identity let \(g=q(D)v\), extended by zero, where \(v\in C_c^\infty(V)\). The second identity in (L12) gives \(S_Vg=v\), and therefore \((I+R_V)g=Pv\). The inverse on \(L^p(V)\) implies \(B_VPv=g\), and hence \(EPv=v\). This checks the same \(E\) on both stated domains. It asserts neither a boundary condition for \(Ef\) nor uniqueness among all local solutions. \(\square\)

## 5. The Hölder alternative, with its scaling made explicit

For \(0<\gamma<1\), let \(\overline C^\gamma(V)\) be the bounded functions with finite seminorm
\([f]_{\gamma,V}=\sup_{x\ne y\in V}|f(x)-f(y)|/|x-y|^\gamma\).
On a ball \(B_r\) use the equivalent, scaled norm
\[
N_r(f)=\|f\|_{\infty,B_r}+r^\gamma[f]_{\gamma,B_r}.                 \tag{L21}
\]
This is a Banach norm: a Cauchy sequence converges uniformly, and the difference quotient bound passes to that limit, including the Cauchy bound for the seminorm. The product inequality
\(N_r(fg)\leq N_r(f)N_r(g)\) follows by adding and subtracting \(f(x)g(y)\).

**Extension.** A function in this space has a unique continuous extension to \(\overline B_r\), by its Hölder bound. Extend it to all of \(\mathbb R^n\) by
\[
\mathcal A_rf(x)=
\begin{cases}
f(x),&|x|\leq r,\\
(2-|x|/r)f(rx/|x|),&r<|x|<2r,\\
0,&|x|\geq2r.
\end{cases}                                                       \tag{L22}
\]
To check the norm including pairs in different regions, let \(\pi_r\) be radial projection onto \(\overline B_r\), and let \(\theta_r(x)=\min(1,\max(0,2-|x|/r))\). Then \(\mathcal A_rf=\theta_r(f\circ\pi_r)\). The projection has Lipschitz constant at most two: for exterior points expand the difference of their normalized vectors, ordering their radii; for an interior and an exterior point insert the point where their connecting segment first meets the sphere. Also
\(|\theta_r(x)-\theta_r(y)|\leq\min(1,|x-y|/r)\leq(|x-y|/r)^\gamma\).
The product difference estimate now gives
\[
\|\mathcal A_rf\|_\infty+r^\gamma[\mathcal A_rf]_{\gamma,\mathbb R^n}
\leq C_\gamma N_r(f),                                             \tag{L23}
\]
with a constant independent of \(r\).

**The frozen Hölder bound.** For \(h\) bounded and globally \(\gamma\)-Hölder, the top-order kernels in (L6) have zero integral. Thus
\[
K_\ell*h(x)=\int K_\ell(z)(h(x-z)-h(x))\,dz,
\quad
\|K_\ell*h\|_\infty\leq C2^{-\ell\gamma}[h]_\gamma.                \tag{L24}
\]
The derivative kernel also has zero integral, and the same computation gives
\(\|\nabla(K_\ell*h)\|_\infty\leq C2^{\ell(1-\gamma)}[h]_\gamma\).
For points a distance \(t\leq1\) apart, estimate a term either by twice its supremum or by \(t\) times its derivative supremum. Sum the derivative bound where \(2^\ell t\leq1\) and the supremum bound where \(2^\ell t>1\). The geometric sums are at most
\(C_\gamma t^\gamma[h]_\gamma\).
For \(t>1\), the convergent supremum sum suffices. Therefore the sum defines a bounded \(C^\gamma\) function with norm at most \(C_\gamma(\|h\|_\infty+[h]_\gamma)\). For compactly supported \(h\), it is the distribution \((D^\alpha F_0)*h\): the symbol partial sums converge on Schwartz test functions, and (L24) identifies their locally uniform limit. For lower orders \(k>0\), the sum of the \(L^1\) kernel norms is finite, so both the supremum and the Hölder seminorm are bounded directly by convolution. Finally, convolution with \(D^\alpha H\), for \(h\) supported in \(B_2\) and output restricted to \(B_1\), is bounded in the full \(C^\gamma\) norm. Use the supremum of that derivative and its first derivatives on \(B_3\), the mean-value theorem, and \(\|h\|_1\leq|B_2|\|h\|_\infty\).

It follows that the operator \(T f=(G*\mathcal A_1 f)|_{B_1}\) satisfies
\[
\|D^\alpha Tf\|_{\overline C^\gamma(B_1)}
\leq C\|f\|_{\overline C^\gamma(B_1)},\qquad |\alpha|\leq m.       \tag{L25}
\]
This argument works directly for the full Hölder space. It does not require smooth functions to be dense in the same Hölder norm.

Rescale this single solver by
\[
S_rf(x)=r^m\,T(f(r\,\cdot))(x/r),\qquad x\in B_r.
\]
Homogeneity of \(q\) gives
\[
q(D)S_rf=f,\qquad
N_r(D^\alpha S_rf)\leq C r^{m-|\alpha|}N_r(f).                    \tag{L26}
\]
If \(g=q(D)v\) with \(v\in C_c^\infty(B_r)\), its rescaled extension in (L22) is precisely its zero extension. Equivalently \(S_rg\) is convolution with the fundamental solution \(r^{m-n}G(\cdot/r)\). Hence \(S_rq(D)v=v\). This remains true when \(G\) has logarithmic terms; no homogeneity assertion about \(G\) was used.

**Theorem.** If every coefficient of \(P\) is \(C^\gamma\) near zero and its frozen principal symbol is elliptic, then on a sufficiently small centered ball there is a linear \(E_\gamma\) on \(\overline C^\gamma\) with
\[
PE_\gamma f=f,\qquad E_\gamma Pv=v\ (v\in C_c^\infty),\qquad
D^\alpha E_\gamma:\overline C^\gamma\to\overline C^\gamma
\text{ bounded for }|\alpha|\leq m.                              \tag{L27}
\]

**Proof.** Compute \(PS_r=I+R_r\) as in (L18). For a highest-order coefficient,
\[
N_r(a_\alpha-a_\alpha(0))\leq2r^\gamma[a_\alpha]_{\gamma,B_r}.
\]
For a lower-order coefficient its norm \(N_r(a_\alpha)\) remains bounded as \(r\downarrow0\), while (L26) supplies the extra factor \(r^{m-|\alpha|}\). The product inequality therefore gives
\[
\|R_r\|_{N_r\to N_r}\leq C\left(
r^\gamma\sum_{|\alpha|=m}[a_\alpha]_{\gamma,B_r}
 +\sum_{|\alpha|<m}r^{m-|\alpha|}N_r(a_\alpha)\right)\longrightarrow0.
                                                                    \tag{L28}
\]
Choose this norm below \(1/2\), and set \(E_\gamma=S_r(I+R_r)^{-1}\). The same geometric-series and test-function argument used for (L20) proves both identities. Equation (L26) gives the stronger estimate
\(N_r(D^\alpha E_\gamma f)\leq2C r^{m-|\alpha|}N_r(f)\).
At fixed \(r\), scaled and unscaled norms are equivalent, proving (L27). The small factor in the top terms is \(r^\gamma\); the unscaled Hölder seminorm of a coefficient need not decrease on smaller balls. \(\square\)

## 6. An estimate uniform over open subsets

In this part the operator is the principal part \(P_m\) alone. Fix a compact neighborhood \(K\) of zero, contained in the coefficient domain, on which the coefficients are continuous and
\[
\left|\sum_{|\alpha|=m}a_\alpha(x)\xi^\alpha\right|
\geq c_K|\xi|^m\qquad(x\in K,\ \xi\in\mathbb R^n).                \tag{L29}
\]
Such a \(K\) is obtained by shrinking around an elliptic point, using continuity and compactness of the unit sphere. For any open \(V\subset K\), let
\(d_V(x)=\operatorname{dist}(x,\mathbb R^n\setminus V)\).

**Theorem.** There is a constant \(C\), depending on \(K,P_m,n,m,p\) but not on \(V\), such that, for \(u\in W^{m,p}(V)\) and \(0\leq j\leq m\),

\[
\|d_V^j\nabla^j u\|_{L^p(V)}
\leq C\bigl(\|d_V^mP_mu\|_{L^p(V)}+\|u\|_{L^p(V)}\bigr)^{j/m}
        \|u\|_{L^p(V)}^{1-j/m}.                                  \tag{L30}
\]

For \(j=m\) this is a linear estimate, and for \(j=0\) it is the identity bound. If \(u=0\) as an \(L^p\) function, every derivative is zero, and the assertion is interpreted accordingly. Lower-order terms are not included in \(P_m\) in (L30).

**Proof.** The frozen polynomials
\(q_y(\xi)=\sum_{|\alpha|=m}a_\alpha(y)\xi^\alpha\), \(y\in K\), form a compact family satisfying (L29). Thus (L13) has a common constant. Let \(\eta\in C_c^\infty(B_1)\) equal one on \(B_{1/2}\). For a ball \(B(y,r)\subset V\), apply (L13) to \(\eta((x-y)/r)u(x)\), extended by zero, with \(A=M/r\). On \(B(y,r/2)\) its derivatives equal those of \(u\). Expanding the frozen operator gives three types of terms:
\[
q_y(D)(\eta_yu)=\eta_yP_mu+\eta_y(q_y(D)-P_m)u+[q_y(D),\eta_y]u.
\]
The middle term is at most \(\omega_a(r)|\nabla^m u|\), up to a fixed array constant, where \(\omega_a(r)\to0\) is a common modulus of continuity of the finitely many coefficients on \(K\). Every term in the last commutator contains a derivative of \(u\) of some order \(j<m\), multiplied by a coefficient bounded by \(Cr^{j-m}\). Taking \(p\)th powers, using the finite-sum inequality, and multiplying by \(r^{mp}\), gives
\[
\begin{split}
\sum_{j=0}^m M^{p(m-j)}r^{pj}
       \int_{B(y,r/2)}|\nabla^j u|^p
\leq C\bigg(&\int_{B(y,r)}|r^mP_mu|^p
 +\omega_a(r)^p\int_{B(y,r)}|r^m\nabla^m u|^p\\
 &+\sum_{j<m}\int_{B(y,r)}|r^j\nabla^j u|^p
 +M^{mp}\int_{B(y,r)}|u|^p\bigg).
\end{split}                                                       \tag{L31}
\]
All constants in this inequality are independent of \(M,r,y,V\).

Choose a small number \(r_0>0\), to be fixed below, and put
\[
\rho(y)=\min(r_0,d_V(y)/2).                                       \tag{L32}
\]
The distance function is 1-Lipschitz, by the triangle inequality and taking an infimum. Thus \(\rho\) is \(1/2\)-Lipschitz. Every ball \(B(y,\rho(y))\) is contained in \(V\). The following two geometric bounds ensure uniform overlap when the centers, rather than a discrete covering, are integrated:
\[
\begin{array}{ll}
|x-y|<\rho(y)&\Longrightarrow\quad
       \rho(y)/2<\rho(x)<3\rho(y)/2,\\[2pt]
|x-y|<2\rho(x)/5&\Longrightarrow\quad
       4\rho(x)/5<\rho(y)<6\rho(x)/5
       \quad\hbox{and}\quad |x-y|<\rho(y)/2.
\end{array}                                                       \tag{L33}
\]
They follow immediately by subtracting the two radii and using the Lipschitz bound. The second ball of centers lies in \(V\), since \(2\rho(x)/5<d_V(x)\). Writing \(v_n=|B_1|\), they imply
\[
\int_{\{y:x\in B(y,\rho(y))\}}\frac{dy}{\rho(y)^n}\leq3^n v_n,
\qquad
\int_{\{y:x\in B(y,\rho(y)/2)\}}\frac{dy}{\rho(y)^n}\geq3^{-n}v_n.
                                                                    \tag{L34}
\]
For the upper bound, the centers lie in \(B(x,2\rho(x))\) and \(\rho(y)>2\rho(x)/3\). For the lower bound, integrate just over \(B(x,2\rho(x)/5)\), where \(\rho(y)<6\rho(x)/5\). The same estimates, with constants depending also on \(j,p\), hold if the integrands are multiplied by \(\rho(y)^{pj}\), after factoring out \(\rho(x)^{pj}\).

Integrate (L31), with \(r=\rho(y)\), against \(dy/\rho(y)^n\). Tonelli and (L34) give
\[
\begin{split}
\sum_{j=0}^m M^{p(m-j)}\|\rho^j\nabla^j u\|_p^p
\leq C\bigg(&\|\rho^mP_mu\|_p^p
 +\omega_a(r_0)^p\|\rho^m\nabla^m u\|_p^p\\
 &+\sum_{j<m}\|\rho^j\nabla^j u\|_p^p+M^{mp}\|u\|_p^p\bigg).
\end{split}                                                       \tag{L35}
\]
First choose \(r_0\) so the coefficient of the highest-order term on the right is less than \(1/2\). Next choose \(M_0\geq1\) sufficiently large that the terms of all lower orders on the right can be absorbed by half of their terms on the left whenever \(M\geq M_0\). Indeed each corresponding left coefficient is at least \(M^p\). This proves, with \(F=\|\rho^mP_mu\|_p\) and \(U=\|u\|_p\),
\[
\|\rho^j\nabla^j u\|_p
\leq C\bigl(M^{j-m}F+M^jU\bigr),\qquad M\geq M_0.               \tag{L36}
\]

If \(U>0\) and \(F>M_0^mU\), choose \(M=(F/U)^{1/m}\), obtaining
\(C F^{j/m}U^{1-j/m}\). If \(F\leq M_0^mU\), choose \(M=M_0\), obtaining \(CU\). In both cases the bound is at most
\(C(F+U)^{j/m}U^{1-j/m}\). Finally \(d_V\) is bounded above by a constant depending only on the bounded set \(K\); (L32) therefore gives \(c\,d_V\leq\rho\leq d_V/2\), with \(c>0\) depending only on \(K,r_0\). Replacing \(\rho\) by \(d_V\) proves (L30), with a constant independent of the shape, connectedness, or boundary regularity of \(V\). \(\square\)

## 7. A Lipschitz coefficient through a mollifier

We now give the exact weak product used in the remainder of the lesson. If \(v\in L^p_{\rm loc}\) and \(a\) is Lipschitz, define
\[
aD_jv=D_j(av)-(D_ja)v                                             \tag{L37}
\]
as a distribution. The two products on the right are locally integrable. This definition agrees with the usual product for smooth \(v\), and depends continuously on \(v\) in \(L^p\) on compact sets, when tested against a fixed compactly supported smooth function.

If \(u\in W^{m-1,p}_{\rm loc}\), choose \(j\) with \(\alpha_j>0\) and use (L37) with \(v=D^{\alpha-e_j}u\) to define \(aD^\alpha u\), \(|\alpha|=m\). This is independent of that choice: smooth approximation of \(u\) in \(W^{m-1,p}\) on a neighborhood of a test function's support makes each definition converge to the same classical product on the approximants. The continuous dependence just proved identifies the limits.

**Commutator lemma.** Suppose \(a:\mathbb R^n\to\mathbb C\) has Lipschitz constant \(M\), \(v\in L^p(\mathbb R^n)\), and \(\varphi\in C_c^\infty(\mathbb R^n)\). Put \(\varphi_\varepsilon(x)=\varepsilon^{-n}\varphi(x/\varepsilon)\). Then
\[
\begin{split}
C_\varepsilon^a v&=(aD_jv)*\varphi_\varepsilon
                     -a(D_jv*\varphi_\varepsilon),\\
\|C_\varepsilon^a v\|_p
&\leq M\|v\|_p\int_{\mathbb R^n}
       \bigl(|\varphi(z)|+|z|\,|D_j\varphi(z)|\bigr)\,dz,
\end{split}                                                       \tag{L38}
\]
and \(C_\varepsilon^a v\to0\) in \(L^p\) for each fixed \(v\).

**Proof.** Applying (L37), or integrating by parts first for smooth \(v\), gives
\[
\begin{split}
C_\varepsilon^a v(x)
={}&\int [a(x-y)-a(x)]v(x-y)D_j\varphi_\varepsilon(y)\,dy\\
 &-\int (D_ja)(x-y)v(x-y)\varphi_\varepsilon(y)\,dy.
\end{split}                \tag{L39}
\]
The right side makes sense for \(v\in L^p\). Its absolute value is bounded by convolution of \(M|v|\) with
\(|y||D_j\varphi_\varepsilon(y)|+|\varphi_\varepsilon(y)|\).
Young's inequality yields (L38), because changing \(y=\varepsilon z\) makes that kernel's \(L^1\) norm independent of \(\varepsilon\). Approximation in \(L^p\) proves that (L39) represents the distribution in (L38), using the continuity of the product (L37) on compact sets. No boundedness of \(a\) on all of \(\mathbb R^n\) is required; its at most linear growth also makes the original products legitimate distributions.

For \(h\in C_c^\infty\), use the unintegrated expression
\[
C_\varepsilon^a h(x)=\int[a(x-y)-a(x)]D_jh(x-y)\varphi_\varepsilon(y)\,dy.
\]
Its \(L^p\) norm is at most
\(\varepsilon M\|D_jh\|_p\int|z||\varphi(z)|\,dz\), which tends to zero. Given \(v\), approximate it by such an \(h\); (L38) bounds \(C_\varepsilon^a(v-h)\) uniformly in \(\varepsilon\). Let \(\varepsilon\downarrow0\) first, and then \(\|v-h\|_p\downarrow0\). This proves strong convergence. \(\square\)

## 8. Recovering the highest derivatives from the equation

**Weak regularity and weighted estimate.** Retain the fixed compact neighborhood and ellipticity of Section 6, and suppose that all coefficients of \(P_m\) are Lipschitz on a neighborhood of \(K\). Let \(V\subset K\) be open. If
\[
u\in W^{m-1,p}(V),\qquad P_mu\in L^p(V),                           \tag{L40}
\]
where \(P_mu\) is defined by (L37), then \(u\in W^{m,p}_{\rm loc}(V)\), all the weighted derivatives in (L30) belong to \(L^p(V)\), and (L30) holds with the same type of constant. Global unweighted highest-derivative integrability up to \(\partial V\) is not asserted.

**Proof.** Fix a ball \(B_0\) compactly contained in \(V\). Choose nested slightly larger balls inside \(V\), a cutoff \(\chi_0\) equal to one near \(\overline B_0\), and a cutoff \(\chi_1\) equal to one on an open neighborhood \(W\) of \(\operatorname{supp}\chi_0\). Take \(\overline W\subset V\). Set \(v=\chi_0u\), extend it by zero, and set \(b_\alpha=\chi_1a_\alpha\), extended by zero to \(\mathbb R^n\). These \(b_\alpha\) are globally Lipschitz and bounded. Product differentiation, justified by the smooth approximation used after (L37), gives
\[
P_mv=\chi_0P_mu+
\sum_{|\alpha|=m}\sum_{\substack{\beta\leq\alpha\\|\beta|<m}}
 \binom{\alpha}{\beta}a_\alpha(D^{\alpha-\beta}\chi_0)D^\beta u.
                                                                    \tag{L41}
\]
Every term on the right is \(L^p\) and supported in \(V\); its zero extension is also \(\sum b_\alpha D^\alpha v\) on the whole space.

Choose \(\varphi\in C_c^\infty\) of integral one and let \(v_\varepsilon=v*\varphi_\varepsilon\). For each top-order \(\alpha\), choose \(j\) as in (L37). Applying (L38) to \(D^{\alpha-e_j}v\in L^p\) shows
\[
\sum_{|\alpha|=m} b_\alpha D^\alpha v_\varepsilon
=(P_mv)*\varphi_\varepsilon
 -\sum_{|\alpha|=m}C_\varepsilon^{b_\alpha}(D^{\alpha-e_j}v)
\longrightarrow P_mv\quad\hbox{in }L^p(\mathbb R^n).              \tag{L42}
\]
For small \(\varepsilon\), the support of \(v_\varepsilon\) is in \(W\), where \(b_\alpha=a_\alpha\). In particular, (L42) is convergence of \(P_mv_\varepsilon\) in \(L^p(W)\). Also \(v_\varepsilon\to v\) in \(L^p\). Apply (L30), with \(j=m\) and open set \(W\), to the smooth difference \(v_\varepsilon-v_\delta\). Its right side tends to zero. Since \(d_W\) has a positive lower bound on \(\overline B_0\), every highest derivative is Cauchy in \(L^p(B_0)\). Its limit is the corresponding distributional derivative of \(v=u\) there. This proves local \(W^{m,p}\) regularity.

To pass to the full open set without imposing boundary regularity, put
\(V_t=\{x\in V:d_V(x)>t\}\), \(t>0\). Its closure is compact in \(V\), since \(V\subset K\) is bounded. Local regularity and a finite compact cover imply \(u\in W^{m,p}(V_t)\). For \(x\in V_t\), its own distance function satisfies
\[
d_V(x)-t\leq d_{V_t}(x)\leq d_V(x).                               \tag{L43}
\]
The first inequality follows because the ball of radius \(d_V(x)-t\) about \(x\) lies in \(V_t\), by the Lipschitz property of \(d_V\); the second follows from \(V_t\subset V\). Hence, as \(t\downarrow0\), the distances, extended by zero off \(V_t\), increase pointwise to \(d_V\). Apply (L30) on \(V_t\). Its constants are independent of \(t\). Monotone convergence for the derivative integrals, and dominated convergence for the \(u\) and \(P_mu\) integrals, give (L30) on \(V\). This also proves the asserted weighted integrability. \(\square\)

## 9. Annular decay and infinite-order vanishing

Let \(P_m\) be elliptic with continuous principal coefficients on a fixed neighborhood of zero. Suppose \(u\in W^{m,p}_{\rm loc}(K^\circ\setminus\{0\})\), and for some real \(N\) and all sufficiently small \(r>0\),
\[
\int_{r<|x|<2r}|u|^p\,dx=O(r^N),\qquad
|P_mu(x)|\leq C\sum_{|\alpha|<m}|x|^{|\alpha|-m}|D^\alpha u(x)|
\quad\hbox{a.e. off zero}.                                      \tag{L44}
\]

**Corollary.** For every \(|\alpha|\leq m\),
\[
\int_{r<|x|<2r}|r^{|\alpha|}D^\alpha u|^p\,dx=O(r^N).             \tag{L45}
\]
The same conclusion holds when the principal coefficients are Lipschitz and the initial assumption is only \(u\in W^{m-1,p}_{\rm loc}\) off zero, with (L44) interpreted distributionally as in (L37). In that case the inequality in (L44) means that the distribution \(P_mu\) has the indicated locally \(L^p\) representative.

**Proof.** On \(A_r=\{r<|x|<2r\}\), let \(d=d_{A_r}\), \(U=\|u\|_{L^p(A_r)}\), and
\(S=\sum_{j<m}\|d^j\nabla^j u\|_{L^p(A_r)}\).
The weak version is first legitimate on this annulus by Section 8; all lower derivatives and \(P_mu\) are \(L^p\) on it because its closure lies in the punctured neighborhood. The same is true in the original strong version. Since \(d\leq r\leq|x|\), (L44) implies
\(F=\|d^mP_mu\|_p\leq C S\). Notice also that \(U\leq S\). Applying (L30) at every \(j<m\),
\[
S\leq C\sum_{j<m}S^{j/m}U^{1-j/m}
\leq C S^{(m-1)/m}U^{1/m}.                                      \tag{L46}
\]
If \(S=0\), the desired bound is immediate. Otherwise divide by \(S^{(m-1)/m}\) and raise to the \(m\)th power, obtaining \(S\leq CU\). For \(m=1\), the sum is just \(S=U\), consistent with this calculation. The top-order case of (L30) then gives \(\|d^m\nabla^m u\|_p\leq C(F+U)\leq CU\).

On \(M_r=\{4r/3<|x|<5r/3\}\) one has \(d\geq r/3\). Thus
\[
\int_{M_r}|r^j\nabla^j u|^p\,dx\leq C U^p=O(r^N),\qquad j\leq m.
                                                                    \tag{L47}
\]
To obtain the original full annulus, not just its middle, use \(M_{s r}\) for
\(s=3/4,9/10,11/10,13/10\). Their radial intervals overlap and cover \((r,2r)\): after dividing by \(r\), they are \((1,5/4)\), \((6/5,3/2)\), \((22/15,11/6)\), and \((26/15,13/6)\). Their associated larger annuli still lie in the coefficient neighborhood for small \(r\). Each application of (L47) has \((sr)^N=O(r^N)\), and \(r^j\) is a fixed multiple of \((sr)^j\). Summing these four bounds proves (L45). \(\square\)

Under the differential inequality in (L44), the condition that its annular integral be \(O(r^N)\) for every \(N>0\) is therefore equivalent to the same condition for every scaled derivative in (L45). It suffices to include orders below \(m\), because they include \(u\); the top-order bounds then follow. This is the infinite-order vanishing statement used later in unique continuation.

The annular and ball formulations agree as well. If an annular bound holds for a positive exponent \(N\), split \(B_r\setminus\{0\}\) into the disjoint annuli of inner radii \(2^{-\nu-1}r\), \(\nu\geq0\). Their bounds sum to at most \(C r^N\sum_{\nu\geq0}2^{-(\nu+1)N}\). The value at the origin has no effect on the integral. Conversely, an annulus is contained in \(B_{2r}\). For an unweighted derivative of order \(j\), replace \(N\) in (L45) by \(N+pj\); this gives every desired positive decay exponent for that derivative as well. This argument establishes decay and integrability; a conclusion that the solution vanishes in a neighborhood requires a unique-continuation theorem in addition.

## 10. Problems with full solutions

**Problem 1: all three coefficient regimes in one operator.** In dimension six, take \(m=4\), \(p=2\), and
\[
P=(1+|x|^{1/3})(D_1^2+\cdots+D_6^2)^2
  +\sum_{|\alpha|\leq3}b_\alpha(x)D^\alpha.
\]
Determine sufficient local spaces for each \(b_\alpha\), and all admissible derivative exponents for its inverse.

**Solution.** The frozen symbol is \(|\xi|^4\); the principal coefficients are continuous, although no first derivative at zero is required. The critical gap is \(n/p=3\). Order-three coefficients have gap one and may lie in \(L^6_{\rm loc}\); order-two coefficients may lie in \(L^3_{\rm loc}\); order-one coefficients require \(L^{2+\varepsilon_\alpha}_{\rm loc}\); and order-zero coefficients require only \(L^2_{\rm loc}\). These coefficients can be unbounded. For example, a coefficient of order three equal to \(|x|^{-1/2}\) near zero is in \(L^6_{\rm loc}(\mathbb R^6)\), because its sixth power has radial integral proportional to \(\int_0^r t^2\,dt\).

On a sufficiently small neighborhood, (L17) gives order-four derivatives in \(L^2\), order-three derivatives in every \(L^q\) with \(2\leq q\leq3\), order-two derivatives for \(2\leq q\leq6\), order-one derivatives for every finite \(q\geq2\), and the solution itself in \(L^\infty\) as well as all these finite spaces. The first-order term uses, in particular, \(q=2(2+\varepsilon_\alpha)/\varepsilon_\alpha\). There is no \(L^\infty\) assertion for the first derivatives at this critical gap.

**Problem 2: why \(L^p\) alone is insufficient in the critical multiplication step.** In a small disk in \(\mathbb R^2\), construct \(a\in L^2\) and \(w\in W^{1,2}\) such that \(aw\notin L^2\). Relate this to the critical row of the table.

**Solution.** Near zero write \(L(r)=\log(e/r)\) and take
\[
w(x)=L(|x|)^{1/4},\qquad
a(x)=|x|^{-1}L(|x|)^{-5/8},
\]
with smooth cutoffs away from zero. The integral for \(|a|^2\) near zero is a constant times \(\int_0^r t^{-1}L(t)^{-5/4}\,dt\), which is finite. The function \(w\) is square-integrable, and its gradient has squared radial integral bounded by \(C\int_0^r t^{-1}L(t)^{-3/2}\,dt<\infty\). These classical derivatives off zero are also its weak derivatives: integration by parts outside a disk of radius \(\delta\) produces a boundary term bounded by \(C\delta L(\delta)^{1/4}\), tending to zero. Thus \(w\in W^{1,2}\). But \(|aw|^2\) has radial integral \(\int_0^r t^{-1}L(t)^{-3/4}\,dt=\infty\).

Here \(k=1=n/p\). The example disproves the unrestricted multiplication estimate \(L^2\cdot W^{1,2}\subset L^2\), which would be needed to use just \(a\in L^p\) at the critical gap. It does not purport to be a nonexistence example for every elliptic equation. The stronger coefficient space \(L^{2+\varepsilon}\), paired with a sufficiently large finite derivative exponent, is exactly what makes (L19) valid. The displayed \(a\) belongs to no \(L^{2+\varepsilon}\), as its extra radial power already forces divergence.

**Problem 3: check the annular weights on a singular solution profile.** Let \(P_2=\sum_jD_j^2=-\Delta\), \(n\geq2\), and \(u(x)=|x|^\lambda\) off zero, with \(\lambda\in\mathbb R\). Find the annular exponent and verify the differential inequality required in (L44).

**Solution.** Direct radial differentiation gives
\(-\Delta u=-\lambda(\lambda+n-2)|x|^{\lambda-2}\).
Thus \(|P_2u|\leq C|x|^{-2}|u|\), one of the allowed terms in (L44). All derivatives exist and are locally \(L^p\) in the punctured neighborhood. On an annulus, changing variables \(x=ry\) gives
\[
\int_{r<|x|<2r}|u|^p\,dx
=r^{p\lambda+n}\int_{1<|y|<2}|y|^{p\lambda}\,dy.
\]
So \(N=p\lambda+n\), which may be negative, zero, or positive. A derivative of order \(j\leq2\) is homogeneous of degree \(\lambda-j\), unless it vanishes identically. Multiplication by \(r^j\) restores the same power \(r^{p\lambda+n}\) after integration. This verifies both the weight and the exponent in (L45), including profiles that are not integrable at the origin before imposing a sufficiently positive exponent.

**Problem 4: strong commutator convergence need not be operator norm convergence.** Let \(a(x)=x_j\) in (L38), and choose \(\varphi\in C_c^\infty\) of integral one. Show why the commutator tends to zero on each fixed \(L^p\) function even though its \(L^p\) operator norm does not tend to zero.

**Solution.** Formula (L39), with \(D_j=-i\partial_j\), gives convolution with
\[
k_\varepsilon(y)=i\partial_{y_j}\bigl(y_j\varphi_\varepsilon(y)\bigr)
=\varepsilon^{-n}k_1(y/\varepsilon).
\]
This is a nonzero smooth compactly supported kernel: if \(\partial_j(y_j\varphi)=0\), then on each line parallel to the \(j\)th axis the compactly supported function \(y_j\varphi\) is constant, hence zero, forcing \(\varphi=0\), contrary to its integral. There is \(h\in C_c^\infty\) with \(k_1*h\ne0\); for instance, convolution with a smooth approximate identity converges to the nonzero \(k_1\). Define \(h_\varepsilon(x)=\varepsilon^{-n/p}h(x/\varepsilon)\). A change of variables gives
\[
\|h_\varepsilon\|_p=\|h\|_p,\qquad
\|k_\varepsilon*h_\varepsilon\|_p=\|k_1*h\|_p>0.
\]
Thus the operator norm is bounded below independently of \(\varepsilon\). The lemma applies to a fixed input; this example changes the input with the smoothing scale. The weak regularity proof in (L42) uses precisely the strong convergence on its finitely many fixed lower derivatives.

## 11. The weak derivative of the original Lipschitz coefficient

The entry fact about Lipschitz coefficients needs a proof because it is used inside (L37), rather than only as a bound on a smooth coefficient. We prove it from the integration results in Sections 15.1–15.5 of [Banach estimates, quotient spaces and compact parameter arguments](banach-foundation-bridges.md) and the Hilbert representation proof in Section 2.3 of [The full calculus of a lower-bounded self-adjoint operator](lower-bounded-spectral-calculus.md). The proof retains the actual complex coefficient. No differentiability almost everywhere or measure-representation theorem for the dual of \(L^1\) is assumed.

### 11.1. Difference quotients and a bounded measurable derivative

Let \(\Omega\subset\mathbb R^n\) be open and let \(a:\Omega\to\mathbb C\) satisfy \(|a(x)-a(y)|\leq M|x-y|\) for all points under consideration, with \(M\geq0\). The local version uses such a bound on each interior neighborhood. Fix a bounded coordinate cube \(Q\) with closure in \(\Omega\). For \(\phi\in C_c^\infty(Q)\), its support and all sufficiently small translates remain in \(Q\). The original change of variables and the scalar fundamental theorem give
\[
 \begin{split}
 -\int_Q a(x)\partial_j\phi(x)\,dx
 &=\lim_{t\to0}\int_Q
       \frac{a(x+t e_j)-a(x)}{t}\phi(x)\,dx,\\
 \left|-\int_Q a\partial_j\phi\right|
 &\leq M\|\phi\|_{L^1(Q)}
 \leq M|Q|^{1/2}\|\phi\|_{L^2(Q)} .
 \end{split}                                                   \tag{LC1}
\]
In the change of variables the difference of translated tests is supported in one fixed compact subset of \(Q\). The function \(a\) is bounded there: compare its values with a fixed point in that neighborhood. The difference quotients of the tests converge uniformly to \(-\partial_j\phi\). Dominated convergence therefore proves the first line, including its minus sign, while the Lipschitz bound proves the second.

Here the dense test subspace of \(L^2(Q)\) can also be verified without a boundary assertion. For an arbitrary \(f\in L^2(Q)\), choose compact smooth cutoffs \(0\leq\chi_k\leq1\) supported in \(Q\), equal to one on increasing interior cubes exhausting \(Q\). Dominated convergence gives \(\chi_k f\to f\) in \(L^2(Q)\). Extend \(\chi_k f\) by zero. Smooth it by the original compact kernel of integral one from (LP14); choosing its scale smaller than the distance of \(\operatorname{supp}\chi_k\) to \(Q^c\), divided by a radius containing the kernel support, keeps the smoothed support inside \(Q\). Formula (LP14) gives arbitrarily small \(L^2\) error. Thus \(C_c^\infty(Q)\) is dense. Every original support and scale restriction is retained.

The linear functional in (LC1) extends uniquely to \(L^2(Q)\). Completeness of that space is (LP10), and its inner product is the original integral \(\langle f,h\rangle=\int_Q f\overline h\), linear in the first variable. The Hilbert representation proof cited above, applied to the conjugate of this linear functional, gives \(g_{j,Q}\in L^2(Q)\) such that
\[
 -\int_Q a\partial_j\phi=\int_Q g_{j,Q}\phi,
 \qquad
 \left|\int_Q g_{j,Q}h\right|\leq M\|h\|_{L^1(Q)}
                         \quad(h\in L^2(Q)).
                                                               \tag{LC2}
\]
The second inequality follows by approximating \(h\) in \(L^2\) by compact tests; their \(L^1\) errors are at most \(|Q|^{1/2}\) times the \(L^2\) errors. The complex representative in (LC2) is the conjugate of the Hilbert representing vector, so no conjugation has been lost.

For measurable \(E\subset Q\), take \(h=1_E\overline{g_{j,Q}}/|g_{j,Q}|\), with value zero where \(g_{j,Q}=0\). It belongs to \(L^2(Q)\), and (LC2) gives \(\int_E|g_{j,Q}|\leq M|E|\). If \(E=\{|g_{j,Q}|>M+\delta\}\) for \(\delta>0\), this forces \(|E|=0\). Taking a countable union over positive reciprocal integers proves
\[
 g_{j,Q}\in L^\infty(Q),\qquad
 \|g_{j,Q}\|_{L^\infty(Q)}\leq M,\qquad
 \partial_j a=g_{j,Q}\quad\hbox{on }Q.                         \tag{LC3}
\]
On overlapping cubes the two representatives agree almost everywhere. Indeed their difference integrates to zero against every compact smooth test in the overlap. Exhaust that overlap by bounded interior cubes and use the density argument just given in each cube. Approximating the complex conjugate of the difference in \(L^2\) proves that its squared modulus has zero integral there.

A countable collection of interior cubes covers \(\Omega\). Define \(g_j\) by taking the first cube containing each point; discard the countable union of the exceptional null sets on overlaps. These measurable representatives agree locally and retain every local bound. To verify the global distribution identity for a compact test, cover its support by finitely many of these cubes, choose nonnegative smooth bumps \(\theta_l\) supported in those cubes whose sum is positive on a neighborhood of the test support, and write \(\phi=\sum_l\phi\theta_l/(\sum_r\theta_r)\). Each summand is a compact smooth test in one cube; all original weights remain in this identity. Summing (LC2) proves \(\partial_j a=g_j\) throughout \(\Omega\). In the original derivative convention, \(D_j a=-i g_j\); hence \(\|D_j a\|_\infty\leq M\) wherever the same Lipschitz bound holds. No global boundedness of \(a\) was required.

### 11.2. The complex gradient bound and the converse on the actual domain

There is a sharper form of the preceding bound, with a necessary distinction for a complex coefficient. Repeat (LC1) in a real direction \(v\in\mathbb Q^n\), replacing \(te_j\) by \(tv\). The resulting functional is represented by \(\sum_j v_jg_j\), by the coordinate identities and uniqueness. Thus \(|\sum_jv_jg_j(x)|\leq M|v|\) almost everywhere. Remove the null exceptional sets for this countable set of directions. Continuity in \(v\) then gives the exact pointwise real linear map
\[
 A(x)v=\sum_{j=1}^n v_jg_j(x),\qquad
 \|A(x):\mathbb R^n\to\mathbb C\|_{\rm op}\leq M
                                      \quad\hbox{a.e.}          \tag{LC4}
\]
For real \(a\), this operator norm is \((\sum_j|g_j|^2)^{1/2}\). For complex \(a\), its real matrix has rows \((\operatorname{Re}g_j)_j\) and \((\operatorname{Im}g_j)_j\). The two row bounds, and the \(n\) coordinate bounds, give both original estimates
\[
 \sum_{j=1}^n|g_j|^2\leq 2M^2,\qquad
 \sum_{j=1}^n|g_j|^2\leq nM^2 .
                                                               \tag{LC5}
\]
For each row the Euclidean norm is the supremum of its real directional pairing, bounded by (LC4). The coordinate estimate is (LC3). The complex Euclidean gradient bound cannot be replaced by \(M\) in general: \(a(x_1,x_2)=x_1+i x_2\) has Lipschitz constant one, original derivative array \((1,i)\), real operator norm one and Euclidean array norm \(\sqrt2\). These are different norms on the same retained derivative array.

Conversely, suppose \(a\in L^1_{\rm loc}(\Omega)\) has weak coordinate derivatives \(g_j\in L^\infty_{\rm loc}\), with the map (LC4) bounded by \(M\) on a neighborhood of a compact segment. Choose a nonnegative compact smooth \(r\) with \(\int r=1\), and retain \(r_\varepsilon(x)=\varepsilon^{-n}r(x/\varepsilon)\). On interior points its local smoothing satisfies \(\partial_j a_\varepsilon=r_\varepsilon*g_j\), by (LP19) after inserting an interior cutoff equal to one near the points and their kernel translates. Consequently, for every real \(v\),
\[
 |\partial_v a_\varepsilon(x)|
 =\left|\int r_\varepsilon(y)A(x-y)v\,dy\right|
 \leq M|v|\int r_\varepsilon(y)\,dy=M|v|.
                                                               \tag{LC6}
\]
The nonnegativity in this converse is the chosen kernel's hypothesis, and its scale and integral remain explicit. Integration along any retained interior segment gives \(|a_\varepsilon(x)-a_\varepsilon(y)|\leq M|x-y|\).

These smoothings converge locally uniformly to a representative of the original \(a\). Here is a direct verification. They are Cauchy in \(L^1\) on every relatively compact interior neighborhood, by (LP14) with an interior cutoff. Fix compact \(K\) and \(\rho>0\) such that its \(\rho\)-neighborhood has compact closure in \(\Omega\). If \(M>0\) and \(|a_\varepsilon(x)-a_\delta(x)|\geq\eta>0\) at \(x\in K\), both smoothings have Lipschitz constant at most \(M\) on a small ball about \(x\). Set \(b=\min(\rho/2,\eta/(4M))\). On the ball of radius \(b\), their difference has modulus at least \(\eta/2\). The coordinate box centered at \(x\) with side lengths \(2b/\sqrt n\) lies in that ball, up to its measure-zero faces. Thus their original \(L^1\) distance on the neighborhood is at least
\[
          \frac{\eta}{2}\left(\frac{2b}{\sqrt n}\right)^n .
                                                               \tag{LC7}
\]
This contradicts the Cauchy property for sufficiently small scales, proving uniform Cauchy convergence on \(K\). If \(M=0\), the smoothings are constant on each such ball, and their \(L^1\) distance is that constant difference times its positive original box volume; the same conclusion follows. Covering \(K\) by finitely many balls handles locally varying bounds. The uniform limit is continuous and is equal to \(a\) almost everywhere by the \(L^1\) limit. Passing to the limit in the segment inequality proves a local Lipschitz representative with exactly the map bound \(M\). On a convex open \(\Omega\) with a global bound, every segment between its points has compact image in \(\Omega\), so the representative is globally \(M\)-Lipschitz. On a general open set, the conclusion holds along interior segments or paths, retaining their actual lengths; no inequality across disconnected components is inferred.

### 11.3. The full weak product, highest derivatives and cutoff extension

Let \(h\) be smooth. Applying (LC2) to \(h\phi\), and using the scalar product rule on that test, gives
\[
 -\int ah\partial_j\phi
       =\int(g_jh+a\partial_jh)\phi,\qquad
 D_j(ah)=(D_j a)h+aD_jh.                                       \tag{LC8}
\]
For \(h\in W^{1,p}_{\rm loc}\), \(1\leq p<\infty\), use its smooth approximations from (LP19) on a neighborhood of the test support. The locally bounded \(a\) and \(g_j\) multiply the converging \(L^p\) components continuously, so (LP16) passes (LC8) to that original \(h\). For \(p=\infty\), take these same local limits in \(L^2\) on finite-measure test neighborhoods; both \(h\) and its derivatives belong there to \(L^2\), while the resulting products retain their local essential bounds. This proves the product identity at that endpoint without a false strong \(L^\infty\) smoothing assertion.

For arbitrary \(v\in L^p_{\rm loc}\), retain precisely the distribution definition (L37). Its full pairing and continuity bound are
\[
 \begin{split}
 \langle aD_jv,\phi\rangle
 &=-\int v aD_j\phi-\int v(D_j a)\phi,\\
 |\langle aD_jv,\phi\rangle|
 &\leq\|v\|_{L^p(K)}
     \left(\|aD_j\phi\|_{L^{p'}(K)}
                    +\|(D_j a)\phi\|_{L^{p'}(K)}\right),
 \qquad K=\operatorname{supp}\phi .
 \end{split}                                                   \tag{LC9}
\]
The conjugate exponents include their two endpoints. Every \(-i\) factor is still inside the original \(D_j\)'s. Formula (LC8) proves agreement with the classical product for smooth \(v\). If \(u\in W^{m-1,p}_{\rm loc}\), \(1\leq p<\infty\), approximate all its derivatives up to \(m-1\) on a neighborhood of \(K\) by one smooth sequence. For each \(j\) with \(\alpha_j>0\), apply (LC9) to \(v=D^{\alpha-e_j}u\). Its approximating products are the same classical \(aD^\alpha u_k\), because the original smooth coordinate derivatives commute. Their distributional limits therefore agree for every choice of \(j\). This proves the claimed independence in (L37), and the same approximations prove the entire product differentiation (L41), retaining each \(\binom{\alpha}{\beta}\) and each original ordered derivative.

The coefficient cutoff extension in Section 8 also has an exact global bound. Let \(\chi\in C_c^\infty(N)\), where \(a\) is Lipschitz with constant \(M\) on the open \(N\). Choose \(\rho>0\) with the compact \(\rho\)-neighborhood of \(\operatorname{supp}\chi\) contained in \(N\), let \(A\) be the supremum of \(|a|\) there, and put \(L_\chi=\sum_j\|\partial_j\chi\|_\infty\). For \(b=\chi a\) extended by zero, the original estimates give
\[
 \|b\|_\infty\leq A\|\chi\|_\infty,\qquad
 \operatorname{Lip}(b)
 \leq\max\left(M\|\chi\|_\infty+A L_\chi,
                     \frac{2A\|\chi\|_\infty}{\rho}\right).
                                                               \tag{LC10}
\]
For points at distance less than \(\rho\), if at least one cutoff value is nonzero, both points lie in that compact neighborhood. Expand the original difference as \(\chi(x)(a(x)-a(y))+a(y)(\chi(x)-\chi(y))\) and use the scalar segment bound for \(\chi\). If neither cutoff value is nonzero, the difference is zero. For points separated by at least \(\rho\), use the two separate amplitude bounds, giving the second constant. This covers points outside \(N\) as well. An identically zero cutoff gives the zero function directly. The full weak derivative is \(D_jb=\chi D_ja+aD_j\chi\), extended by zero: an interior test cutoff equal to one near \(\operatorname{supp}\chi\) proves that no boundary term is added. Thus every original \(b_\alpha=\chi_1a_\alpha\) used in (L41)–(L42) is actually bounded and globally Lipschitz.

### 11.4. The commutator bound with every scale factor

The commutator argument has a useful precise scope: its bound holds for \(1\leq p\leq\infty\), and its strong convergence holds for \(1\leq p<\infty\). This strengthens the commutator itself; the elliptic inverse theorems above keep their original \(1<p<\infty\) hypotheses.

For \(a:\mathbb R^n\to\mathbb C\) globally \(M\)-Lipschitz and \(v\in L^p\), insert (LC9) into the original convolution in (L38). The differentiation signs give exactly (L39); the first integral contains \([a(x-y)-a(x)]D_j\varphi_\varepsilon(y)\), and the second contains \(-(D_ja)(x-y)\varphi_\varepsilon(y)\). The convolution kernels are compact, so each pairing is defined. If \(p\) is finite, the at most linear growth \(|a(x)|\leq|a(0)|+M|x|\) also gives a tempered product on the original Schwartz tests by Hölder with the full weight \((|a(0)|+M|x|)\); the same integral bound applies at infinity. Nothing requires globally bounded \(a\).

Using (LC3) and the exact Lipschitz difference, the absolute kernel is bounded by \(M(|y||D_j\varphi_\varepsilon(y)|+|\varphi_\varepsilon(y)|)\). Its full change of variables is
\[
 \begin{split}
 &\int\left(|y|\,\varepsilon^{-n-1}
                |D_j\varphi(y/\varepsilon)|
        +\varepsilon^{-n}|\varphi(y/\varepsilon)|\right)dy\\
 &=\int\left(\varepsilon|z|\,\varepsilon^{-n-1}|D_j\varphi(z)|
             +\varepsilon^{-n}|\varphi(z)|\right)\varepsilon^n dz\\
 &=\int\left(|z|\,|D_j\varphi(z)|+|\varphi(z)|\right)dz .
 \end{split}                                                   \tag{LC11}
\]
Young's (LP9) proves (L38) at every displayed norm endpoint. For compact smooth \(h\), the unintegrated original commutator has norm at most \(\varepsilon M\|D_jh\|_p\int|z||\varphi(z)|\,dz\). For finite \(p\), choose such \(h\) with arbitrarily small \(\|v-h\|_p\), using Section 15.3 of the integration chapter. The complete estimate is
\[
 \|C_\varepsilon^a v\|_p
 \leq M\|v-h\|_p\int
          (|\varphi(z)|+|z||D_j\varphi(z)|)\,dz
       +\varepsilon M\|D_jh\|_p\int|z||\varphi(z)|\,dz .
                                                               \tag{LC12}
\]
Let the scale tend to zero and then the approximation error tend to zero. This proves the original strong limit needed by every term of (L42), and also the commutator's \(p=1\) extension.

At \(p=\infty\), strong convergence for every input is false. In dimension one take \(a(x)=|x|\), \(v=1_{(0,\infty)}\), and a nonzero compact smooth \(\varphi\). The actual distribution products are \(D v=-i\delta_0\) and \(aD v=0\), as follows either from (LC9) or from \(D(x_+)=-i1_{(0,\infty)}\). Therefore
\[
 C_\varepsilon^a v(x)=i|x|\,\varepsilon^{-1}\varphi(x/\varepsilon),
 \qquad
 \|C_\varepsilon^a v\|_\infty
          =\sup_z |z|\,|\varphi(z)|>0 .                        \tag{LC13}
\]
The point \(z=0\) contributes zero; nonzero smooth \(\varphi\) has a nonzero value at some nonzero point, proving positivity. The original boundedness estimate survives, while its finite-norm approximation argument has exactly the finite-\(p\) scope already proved.

These proofs close the Lipschitz entry used in (L37)–(L42). They supply the actual bounded weak derivative, the product pairing and choice independence, the global coefficient cutoff bound and the scale-independent commutator estimate. They preserve the original complex coefficient, its coordinate derivatives, every mixed derivative and cutoff contribution, and the finite-\(p\) regularity argument. Sections 12–13 supply the fractional-integration and multiplier proofs with their original hypotheses and exact receiving constants.

## 12. Fractional integration with every original scale retained

We now prove the fractional-integration prerequisite in Section 1. The kernel is the original \(|x|^{s-n}\); no constant is absorbed into it and no input norm is set to one. The proof uses the completed coordinate measure, product integration, Hölder, finite-\(p\) translation continuity and monotone convergence from Section 15 of [Banach estimates, quotient spaces and compact parameter arguments](banach-foundation-bridges.md). Its finite-cube compactness is proved in Section 15.0 there. We do not assume an interpolation theorem, a dual representation theorem or a Lorentz-space theorem in this proof.

### 12.1. Cube averages and the full weak bound

For \(r>0\), let \(Q(x,r)=x+(-r,r)^n\), with its original volume \((2r)^n\). For a locally integrable measurable \(f\), define
\[
 \mathcal Mf(x)=\sup_{r>0}\frac1{(2r)^n}
                         \int_{Q(x,r)}|f(y)|\,dy.              \tag{FI1}
\]
Changing the open faces to closed faces does not alter the integral. The average at a fixed \(r\) is continuous in \(x\). To prove this locally, truncate \(f\) to a sufficiently large bounded coordinate cube containing all relevant \(Q(x,r)\), obtaining a global \(L^1\) function \(g\). Write the average as its convolution with the bounded cube indicator. Translation by \(h\) changes its numerator by at most \(\|\tau_hg-g\|_1\), which tends to zero by (LP12). The same argument applies in every bounded neighborhood of \(x\).

For a fixed \(x\), the numerator is continuous in \(r>0\): on a compact radius interval the indicators converge almost everywhere as \(r\) varies, and all integrands are dominated by \(|f|\) on a fixed bounded cube. The only exceptions are coordinate faces, which are null. Dominated convergence (LP2) applies. Consequently the supremum in (FI1) is also the supremum over positive rational \(r\), so it is measurable. Its superlevel sets are open because it is a supremum of continuous functions of \(x\).

Suppose \(f\in L^1(\mathbb R^n)\) and \(\lambda>0\). Every \(x\) in \(E_\lambda=\{\mathcal Mf>\lambda\}\) is the center of an open cube with average greater than \(\lambda\). Fix a compact \(K\subset E_\lambda\) and take a finite subcover from these cubes. The finite-cover property follows from the nested-box proof in Section 15.0: enclose \(K\) in a closed coordinate cube, add its open complement relative to \(K\) to the cover, and apply that proved finite-cover result.

Choose among the finite family a cube of largest radius, retain it, and discard every cube meeting it. Repeat on the undiscarded family. The retained cubes \(Q_j=Q(x_j,r_j)\) are disjoint. Every discarded cube has radius at most that of the retained cube it met. If \(Q(x,r)\) meets \(Q(x_j,r_j)\) and \(r\leq r_j\), then
\(|x-x_j|_\infty<r+r_j\);
for every \(y\in Q(x,r)\),
\(|y-x_j|_\infty<2r+r_j\leq3r_j\).
Thus the discarded cube lies in \(Q(x_j,3r_j)\), with the exact volume
\[
 |Q(x_j,3r_j)|=(6r_j)^n=3^n(2r_j)^n,\qquad
 |K|\leq\sum_j(6r_j)^n
       <\frac{3^n}{\lambda}\sum_j\int_{Q_j}|f|
       \leq\frac{3^n}{\lambda}\|f\|_1.                        \tag{FI2}
\]
No contribution from a discarded cube is counted in the last sum. Disjointness of the retained cubes gives precisely the last inequality.

To pass from compact subsets to the open set, exhaust \(E_\lambda\) by
\(K_k=\{x\in[-k,k]^n:\operatorname{dist}_\infty(x,E_\lambda^c)\geq1/k\}\).
If the complement is empty, take \(K_k=[-k,k]^n\). The distance to a nonempty closed set is continuous: the triangle inequality gives its difference bound by \(|x-x'|_\infty\). Thus each \(K_k\) is closed and bounded and is compact by the same closed-cube proof. The \(K_k\)'s increase to \(E_\lambda\), since every point of an open set has a small interior cube. Measure continuity from (LP1) now proves
\[
 |\{x:\mathcal Mf(x)>\lambda\}|
              \leq\frac{3^n}{\lambda}\|f\|_1.                 \tag{FI3}
\]
In particular \(\mathcal Mf<\infty\) almost everywhere for \(f\in L^1\), by sending \(\lambda\) to infinity in this bound. The operator is subadditive, is unchanged when an input is changed on a null set, and satisfies
\(\mathcal Mf\leq\|f\|_\infty\) for essentially bounded input. These statements follow from the corresponding numerator inequalities in each original cube.

### 12.2. The strong maximal bound without interpolation

Let \(1<p<\infty\), \(f\in L^p\), and \(\lambda>0\). Put
\(f_\lambda=f1_{\{|f|>\lambda/2\}}\).
The pointwise bound
\(|f_\lambda|\leq(\lambda/2)^{1-p}|f|^p\)
makes it \(L^1\), while \(|f-f_\lambda|\leq\lambda/2\) almost everywhere. Subadditivity and (FI3) give
\[
 |\{\mathcal Mf>\lambda\}|
 \leq|\{\mathcal Mf_\lambda>\lambda/2\}|
 \leq\frac{2\,3^n}{\lambda}
                     \int_{\{|f|>\lambda/2\}}|f(x)|\,dx.      \tag{FI4}
\]
For any measurable \(h\geq0\), possibly infinite, the scalar identity
\(h(x)^p=p\int_0^{h(x)}\lambda^{p-1}\,d\lambda\)
and nonnegative (LP4) give the level-set formula
\(\int h^p=p\int_0^\infty\lambda^{p-1}|\{h>\lambda\}|\,d\lambda\).
Use it for \(h=\mathcal Mf\), insert (FI4), and apply nonnegative product integration once more. All original cutoff and power factors give
\[
 \begin{split}
 \|\mathcal Mf\|_p^p
 &\leq 2\,3^n p\int_{\mathbb R^n}|f(x)|
          \left(\int_0^{2|f(x)|}\lambda^{p-2}\,d\lambda\right)dx\\
 &=\frac{2\,3^n p}{p-1}\int_{\mathbb R^n}
                    |f(x)|(2|f(x)|)^{p-1}\,dx
   =\frac{2^p3^n p}{p-1}\|f\|_p^p .
 \end{split}                                                   \tag{FI5}
\]
When \(f(x)=0\), the inner integral is over an empty interval and its contribution is zero. The condition \(p>1\) is exactly what makes the lower endpoint integrable. Define the explicit retained constant
\[
 C^{\mathcal M}_{n,p}=\left(\frac{2^p3^n p}{p-1}\right)^{1/p};
 \qquad \|\mathcal Mf\|_p\leq C^{\mathcal M}_{n,p}\|f\|_p.                 \tag{FI6}
\]
This proof establishes the estimate directly on the complete original \(L^p\) space, with no interpolation assumption. It also makes \(\mathcal Mf\) finite almost everywhere for these inputs.

### 12.3. The original fractional kernel and its finite endpoint

Fix \(0<s<n\) and \(1<p<n/s\). Write
\(p'=p/(p-1)\), \(F=\|f\|_p\), and
\[
 \theta=1-\frac{sp}{n}>0,\qquad
 q=\frac p\theta=\frac{np}{n-sp},\qquad
 \gamma=(n-s)p'-n>0.                                          \tag{FI7}
\]
Thus \(1/q=1/p-s/n\), \(q>p\), and \(\theta q=p\).
The sign of \(\gamma\) follows by multiplying \(p<n/s\) by its positive denominators. For a measurable \(f\in L^p\), consider the nonnegative potential
\[
 I_s|f|(x)=\int_{\mathbb R^n}|x-y|^{s-n}|f(y)|\,dy .
\]
Its integrand at \(y=x\) may be set to zero, as that singleton is null. The function is measurable by completed nonnegative product integration. Local integrability of \(f\) follows from Hölder on each bounded coordinate cube.

For \(R>0\), split the original integral into \(0<|x-y|\leq R\) and \(|x-y|>R\). Decompose the first part into the annuli
\(2^{-j-1}R<|x-y|\leq2^{-j}R\), \(j\geq0\).
Since \(s-n<0\), the kernel on the \(j\)-th annulus is at most
\((2^{-j-1}R)^{s-n}\).
Its input integral is bounded by the integral on \(Q(x,2^{-j}R)\); sphere and cube boundary choices have no effect here. The defining cube average gives
\[
 \begin{split}
 \int_{0<|x-y|\leq R}|x-y|^{s-n}|f(y)|\,dy
 &\leq\sum_{j=0}^\infty
       (2^{-j-1}R)^{s-n}(2\,2^{-j}R)^n\mathcal Mf(x)\\
 &=\frac{2^{2n-s}}{1-2^{-s}}R^s\mathcal Mf(x)
   =A_{n,s}R^s\mathcal Mf(x),\\
 A_{n,s}&=\frac{2^{2n-s}}{1-2^{-s}} .
 \end{split}                                                   \tag{FI8}
\]
To justify the boundary comment without a higher-dimensional polar formula, every fixed sphere is null: for a fixed first \(n-1\) coordinates its last-coordinate section has at most two points, hence one-dimensional measure zero. Nonnegative product integration (LP4) gives zero total measure. When \(n=1\) the sphere is itself at most two points. All the sets and integrands are Borel apart from the input's already treated completed null modification.

For the far part use Hölder, and split the kernel's \(p'\)-th power into annuli \(2^jR<|z|\leq2^{j+1}R\). Each is contained in the coordinate cube of radius \(2^{j+1}R\), whose full volume is \((2^{j+2}R)^n\). Therefore
\[
 \begin{split}
 \int_{|z|>R}|z|^{(s-n)p'}\,dz
 &\leq\sum_{j=0}^\infty
           (2^jR)^{(s-n)p'}(2^{j+2}R)^n
   =\frac{2^{2n}}{1-2^{-\gamma}}R^{-\gamma},\\
 \int_{|x-y|>R}|x-y|^{s-n}|f(y)|\,dy
 &\leq B_{n,s,p}F R^{s-n/p},\\
 B_{n,s,p}
 &=\left(\frac{2^{2n}}{1-2^{-\gamma}}\right)^{1/p'},
 \qquad -\frac{\gamma}{p'}=s-\frac np .
 \end{split}                                                   \tag{FI9}
\]
The last exponent identity retains the calculation
\(-\gamma/p'=-(n-s)+n/p'=s-n/p\).
The convergence of the full geometric series uses the actual \(\gamma>0\).

If \(F=0\), the input vanishes almost everywhere and its potential is zero. Suppose \(F>0\). At a point where \(0<\mathcal Mf(x)<\infty\), choose the actual radius
\(R=(F/\mathcal Mf(x))^{p/n}\).
Combining (FI8)–(FI9) gives
\[
 \begin{split}
 I_s|f|(x)
 &\leq A_{n,s}\mathcal Mf(x)R^s
           +B_{n,s,p}F R^{s-n/p}\\
 &=(A_{n,s}+B_{n,s,p})
                F^{sp/n}(\mathcal Mf(x))^{1-sp/n}.
 \end{split}                                                   \tag{FI10}
\]
At a point where \(\mathcal Mf(x)=0\), the near bound is zero for every \(R\); the far bound tends to zero as \(R\to\infty\), because \(s-n/p<0\). Thus the same conclusion holds there. The remaining points, where the maximal function is infinite, form a null set by (FI6).

Raise (FI10) to the \(q\)-th power and use \(\theta q=p\). The complete bound is
\[
 \begin{split}
 \|I_s|f|\|_q
 &\leq(A_{n,s}+B_{n,s,p})
                 F^{sp/n}\|\mathcal Mf\|_p^\theta\\
 &\leq H_{n,s,p}\|f\|_p,\qquad
 H_{n,s,p}=(A_{n,s}+B_{n,s,p})\bigl(C^{\mathcal M}_{n,p}\bigr)^{\,1-sp/n}.
 \end{split}                                                   \tag{FI11}
\]
Here \(sp/n+\theta=1\) accounts for the final input norm; both geometric-series constants and the whole maximal constant remain explicit.

In particular the original integral for \(I_sf\) converges absolutely almost everywhere for real or complex \(f\), and \(|I_sf|\leq I_s|f|\). Inputs equal almost everywhere give equal output classes: translate the input null set at each fixed \(x\) and use the completed integral, with the kernel value at its singular singleton as specified. Integral linearity gives a linear map \(I_s:L^p\to L^q\), and (FI11) proves its boundedness. This proves the whole original fractional-integration contract, with its actual kernel and finite endpoint.

### 12.4. The weak first endpoint and two proved strong-endpoint obstructions

For \(f\in L^1\), put \(F=\|f\|_1\). The near estimate (FI8) is unchanged. On \(|x-y|>R\), the original decreasing kernel is at most \(R^{s-n}\), so the far estimate is \(FR^{s-n}\). If \(F>0\) and \(0<\mathcal Mf(x)<\infty\), choose \(R=(F/\mathcal Mf(x))^{1/n}\). Treat zero maximal values as above; its infinite values are null by (FI3). With \(\sigma=n/(n-s)>1\), the resulting bound and its level estimate are
\[
 \begin{split}
 I_s|f|(x)&\leq(A_{n,s}+1)F^{s/n}
                          (\mathcal Mf(x))^{1-s/n},\\
 |\{x:|I_sf(x)|>\tau\}|
 &\leq3^n(A_{n,s}+1)^\sigma(F/\tau)^\sigma,
                       \qquad \tau>0 .
 \end{split}                                                   \tag{FI12}
\]
Indeed the level implication requires
\(\mathcal Mf>(\tau/((A_{n,s}+1)F^{s/n}))^\sigma\).
Insert this value into (FI3); the remaining norm exponent is
\(1+s\sigma/n=\sigma\).
The case \(F=0\) gives a zero potential directly. Thus this is a proved weak \(L^1\)-to-\(L^\sigma\) bound, and also proves that the absolutely defined potential is finite almost everywhere for \(L^1\) input.

The strong \(L^\sigma\) assertion at this endpoint is false. Take the original
\(f=1_{[-1,1]^n}\), whose integral is \(2^n\). On
\(2^j<|x|_\infty\leq2^{j+1}\), \(j\geq0\), each \(y\) in its support satisfies
\(|x-y|\leq\sqrt n(2^{j+1}+1)\).
The negative kernel exponent consequently gives
\[
 \begin{split}
 I_sf(x)&\geq 2^n
                  [\sqrt n(2^{j+1}+1)]^{s-n},\\
 \int_{\{2^j<|x|_\infty\leq2^{j+1}\}}|I_sf(x)|^\sigma\,dx
 &\geq(2^n)^\sigma n^{-n/2}
             \frac{(2^{j+2})^n-(2^{j+1})^n}{(2^{j+1}+1)^n}\\
 &\geq(2^n)^\sigma n^{-n/2}\frac{4^n-2^n}{3^n}>0.
 \end{split}                                                   \tag{FI13}
\]
We used \((s-n)\sigma=-n\) and \(2^{j+1}+1\leq3\,2^j\), retaining both shell volumes. The shells are disjoint, so their infinite sum diverges. This disproves the strong first endpoint.

The other excluded assertion is strong \(L^{n/s}\)-to-\(L^\infty\). Put
\(p_0=n/s>1\), choose \(1/p_0<a\leq1\), and keep the original shells and coefficients
\[
 \begin{split}
 A_j&=\{y:2^{-j-1}<|y|_\infty\leq2^{-j}\},\qquad j\geq1,\\
 f(y)&=\sum_{j=1}^\infty j^{-a}2^{js}1_{A_j}(y),\\
 \|f\|_{p_0}^{p_0}
 &=\sum_{j=1}^\infty j^{-ap_0}2^{jsp_0}
             \bigl((2^{1-j})^n-(2^{-j})^n\bigr)
   =(2^n-1)\sum_{j=1}^\infty j^{-ap_0}<\infty .
 \end{split}                                                   \tag{FI14}
\]
The series converges because \(ap_0>1\), by comparison with the elementary integral of \(t^{-ap_0}\). If \(|x|_\infty\leq2^{-J}\) and \(j\leq J\), then
\(|x-y|\leq2\sqrt n\,2^{-j}\) for \(y\in A_j\).
Using \(s-n<0\), the full lower bound is
\[
 \begin{split}
 I_sf(x)
 &\geq\sum_{j=1}^J
      (2\sqrt n\,2^{-j})^{s-n}j^{-a}2^{js}
          \bigl((2^{1-j})^n-(2^{-j})^n\bigr)\\
 &=(2\sqrt n)^{s-n}(2^n-1)\sum_{j=1}^Jj^{-a}.
 \end{split}                                                   \tag{FI15}
\]
The finite sums tend to infinity because \(a\leq1\), again by the elementary integral comparison. Each lower bound holds on a cube of positive measure \((2^{1-J})^n\). Hence the output has infinite essential supremum, rather than merely an infinite value at one exceptional point. Hölder on the supporting cube gives \(\|f\|_1\leq(2\,2^{-1})^{n(1-1/p_0)}\|f\|_{p_0}<\infty\). Its far integral is harmless for this compactly supported \(L^1\) input; local absolute convergence almost everywhere also follows from the weak estimate just proved. Thus it gives an actual measurable potential with unbounded essential supremum. The original elliptic statements still exclude this strong infinity endpoint.

### 12.5. The exact receiving kernel and derivative-array estimates

For (L10), put \(k=m-|\alpha|\) with \(0<k<n/p\), retaining the original frozen polynomial and cutoff. Take the finite original bound \(C_{\alpha,\mathrm{near}}=\sup_{0<|z|\leq1}|z|^{n-k}|D^\alpha F_0(z)|\) from (L8). For an integer \(N>n-k\), use the actual Schwartz bound of its full Fourier integral in (L6). The complete annular estimate (L7) then gives, when \(|z|\geq1\),
\[
 \begin{split}
 C_{\alpha,N}&=\sup_z(1+|z|)^N|\kappa_\alpha(z)|,\\
 \kappa_\alpha(z)&=(2\pi)^{-n}
       \int e^{iz\cdot\xi}\psi(\xi)\frac{\xi^\alpha}{q(\xi)}\,d\xi,\\
 \sum_{\ell=0}^\infty|K_\ell(z)|
 &\leq C_{\alpha,N}|z|^{-N}
                  \sum_{\ell=0}^\infty2^{\ell(n-k-N)}
   =\frac{C_{\alpha,N}}{1-2^{n-k-N}}|z|^{-N}\\
 &\leq\frac{C_{\alpha,N}}{1-2^{n-k-N}}|z|^{k-n},\\
 C_\alpha&=C_{\alpha,\mathrm{near}}
                  +\frac{C_{\alpha,N}}{1-2^{n-k-N}} .
 \end{split}                                                   \tag{FI16}
\]
Thus the original kernel \(D^\alpha F_0\), already represented by its \(L^1\) annular sum for \(k>0\), satisfies
\(|D^\alpha F_0(z)|\leq C_\alpha|z|^{k-n}\)
almost everywhere in the whole original space. The Fourier inverse factor in its definition (L6) remains in \(C_{\alpha,N}\); this comparison does not redefine the kernel or its quantization.

Apply (FI11) with \(s=k\) to the original input \(g\). Absolute domination proves
\[
 \|D^\alpha F_0*g\|_q
 \leq C_\alpha H_{n,k,p}\|g\|_p,\qquad
       \frac1q=\frac1p-\frac kn,\quad 0<k<n/p.                \tag{FI17}
\]
This supplies the finite endpoint in (L10), without interpolation, Lorentz duality or a Fourier convention change. The compactly supported zero extension of an input on \(V\subset B_1\) has exactly its original \(L^p(V)\) norm; the constant is independent of \(V\).

For the finite array \(\nabla^j\) used in Section 1, \(q\geq p\) lets us apply Minkowski to the sum of the \(p\)-th powers in \(L^{q/p}\). If \(k=m-j\) satisfies the same gap condition, every component therefore gives the exact array bound
\[
 \begin{split}
 \left\|\left(\sum_{|\alpha|=j}
             |D^\alpha F_0*g|^p\right)^{1/p}\right\|_q
 &\leq\left(\sum_{|\alpha|=j}
                 \|D^\alpha F_0*g\|_q^p\right)^{1/p}\\
 &\leq H_{n,k,p}
         \left(\sum_{|\alpha|=j}C_\alpha^p\right)^{1/p}\|g\|_p .
 \end{split}                                                   \tag{FI18}
\]
Every original multi-index and component constant is retained. The number of terms is \(\binom{n+j-1}{j}\), as proved after (LP17), with value one when \(j=0\). We do not substitute a single component for this array. For the smooth correction \(H=G-F_0\) in (L11), all original differences \(x-y\), \(x,y\in V\subset B_1\), belong to \(B_2\). Hölder gives the original \(\|g\|_1\leq|B_1|^{1-1/p}\|g\|_{L^p(V)}\); each output derivative is bounded by its actual supremum on \(B_2\) times this input integral. Therefore
\[
 \begin{split}
 \|\nabla^j(H*g)\|_{L^q(V;\ell^p)}
 &\leq |V|^{1/q}|B_1|^{1-1/p}
      \left(\sum_{|\alpha|=j}
          \sup_{z\in B_2}|D^\alpha H(z)|^p\right)^{1/p}
        \|g\|_{L^p(V)}\\
 &\leq |B_1|^{1/q+1-1/p}
      \left(\sum_{|\alpha|=j}
          \sup_{z\in B_2}|D^\alpha H(z)|^p\right)^{1/p}
        \|g\|_{L^p(V)} .
 \end{split}                                                   \tag{FI19}
\]
These suprema are finite because each derivative is continuous on the compact closure of \(B_2\). The array triangle inequality combines this with (FI18), retaining both contributions. The strict Young cases still use (L9). Together these prove the fractional portion of (L11) and its coefficient-table applications at the finite endpoint. Section 13 supplies the top-order multiplier proof and its full receiving constants. The general constant-strength antecedent retains its separate proof work.

## 13. The full finite-exponent multiplier proof

The following proof supplies the original multiplier entry used in Sections 2–4. It keeps its Fourier phases, inverse factor, original symbol and exceptional point. All integration and completeness operations are proved in the linked course chapters.

Fix \(n\geq1\), \(M=n+2\), and the original scalar symbol \(b\in C^M(\mathbb R^n\setminus\{0\})\). Retain
\[
 B=\max_{|\beta|\leq M}\sup_{\xi\ne0}
          |\xi|^{|\beta|}|\partial^\beta b(\xi)|<\infty .
                                                               \tag{MP1}
\]
The value at the exceptional point zero can be chosen arbitrarily: its frequency measure is zero by the completed coordinate measure proof in Section 15.0 of the linked integration chapter, and every transform and multiplier below respects the completed null class. We take it to be zero for the integral representatives. There is no atom at zero in the original Lebesgue frequency measure. The original inverse factor and phases are
\[
 Ff(\xi)=\int e^{-ix\cdot\xi}f(x)\,dx,\qquad
 Gh(x)=(2\pi)^{-n}\int e^{ix\cdot\xi}h(\xi)\,d\xi,\qquad
 Tf=G(bFf).
                                                               \tag{MP2}
\]

### 13.1. Constructing the actual \(L^2\) Fourier maps

Theorems 1.1 and 2.1 of [Fourier transforms, finite spectra and convex separation](prerequisite-bridges.md) prove the inverse identities and
\(\|Ff\|_2=(2\pi)^{n/2}\|f\|_2\) on the original Schwartz space. Compact smooth functions are dense in \(L^2\) in Section 15.3 of [Banach estimates, quotient spaces and compact parameter arguments](banach-foundation-bridges.md), so the Schwartz space is dense too. For each \(f\in L^2\), choose \(f_j\) in that space tending to \(f\). The exact norm identity makes \(Ff_j\) Cauchy in the complete \(L^2\) space; its limit is independent of the approximating sequence by the same difference identity. Define \(Ff\) by this limit. The inverse map extends similarly with norm factor \((2\pi)^{-n/2}\), because \(G\) is the inverse on Schwartz space. The two inverse identities extend by density and continuity. Section 7.1 of the Fourier chapter constructs these maps on their original complete spaces; Section 7.2 proves the simultaneous approximations used next. Thus these are actual bijective Fourier maps on the original \(L^2\) space, with both factors retained.

If \(f\in L^1\cap L^2\), truncation in space and in value followed by the mollification proof there gives a single sequence of compact smooth functions converging in both norms. The ordinary Fourier integrals converge uniformly by their \(L^1\) difference bound, while their \(L^2\) classes converge to the constructed Fourier limit. An almost-everywhere convergent subsequence of the latter, supplied by the summable-difference argument in its completeness proof, identifies that class with the ordinary integral. The same argument for \(G\) retains its extra \((2\pi)^{-n}\) in the uniform difference bound. Polarization of the original Schwartz Parseval identity, followed by \(L^2\) convergence, gives the exact pairing
\[
 \langle u,v\rangle
   =(2\pi)^{-n}\int Fu(\xi)\overline{Fv(\xi)}\,d\xi,\qquad
 \|Tf\|_2\leq B\|f\|_2,\qquad
 T^*f=G(\overline b\,Ff).
                                                               \tag{MP3}
\]
Here inner products are linear in their first variable. The adjoint identity follows directly by substituting the product into the displayed pairing; it does not assume an \(L^p\) dual representation. On Schwartz input, \(bFf\in L^1\cap L^2\), so (MP2) is exactly the absolutely defined original multiplier integral.

### 13.2. Complete annular kernel and gradient constants

Use the original compact smooth cutoff \(\chi\), equal to one for \(|\eta|\leq1/2\) and zero for \(|\eta|\geq1\), and put
\[
 \psi(\eta)=\chi(\eta/2)-\chi(\eta),\qquad
 b_j(\xi)=\psi(2^{-j}\xi)b(\xi),\qquad j\in\mathbb Z .
                                                               \tag{MP4}
\]
The support of \(\psi\) is contained in \(1/2\leq|\eta|\leq2\). No positivity of this cutoff is required. For \(a_j(\eta)=\psi(\eta)b(2^j\eta)\), the inverse integral is
\[
 K_j(x)=(2\pi)^{-n}2^{jn}
                 \int e^{i2^jx\cdot\eta}a_j(\eta)\,d\eta .
                                                               \tag{MP5}
\]
These finite integrals define smooth output functions: every output derivative inserts its actual compact-frequency monomial, which is integrable. We require only \(M\) frequency derivatives, rather than incorrectly asserting that the finite-regularity frequency symbol is Schwartz.

For each coordinate \(i\) and \(0\leq r\leq M\), define the actual cutoff constant
\[
 P_{i,r}=\sum_{t=0}^r\binom rt2^t
                    \|\partial_i^{r-t}\psi\|_1,\qquad
 P_0=\|\psi\|_1 .
                                                               \tag{MP6}
\]
The chain derivative of \(b(2^j\eta)\) is \(2^{jt}(\partial_i^tb)(2^j\eta)\). Its modulus is at most \(B|\eta|^{-t}\leq B2^t\) on the original support. The full Leibniz sum therefore gives
\(\|\partial_i^ra_j\|_1\leq BP_{i,r}\).
Choose a coordinate \(i\) with \(|x_i|\geq|x|/\sqrt n\). Integrating \(M\) times in that original coordinate retains the factor \((i2^jx_i)^{-M}\); compact support eliminates the boundary terms. The direct integral and this integrated integral prove
\[
 \begin{split}
 c_0&=(2\pi)^{-n}
            \max\{P_0,n^{M/2}\max_iP_{i,M}\},\\
 |K_j(x)|&\leq c_0B\,2^{jn}
                     \min\{1,(2^j|x|)^{-M}\}.
 \end{split}                                                   \tag{MP7}
\]
All constants are finite and depend only on the actual cutoff and dimension. On \(|x|\leq2^{-j}\), the original containing cube has volume \((2^{1-j})^n\). On \(2^{r-j}<|x|\leq2^{r-j+1}\), \(r\geq0\), use its containing cube of volume \((2^{r-j+2})^n\) and the second bound in (MP7). The full integral comparison gives
\[
 \|K_j\|_1\leq c_0B\,2^{jn}(2^{1-j})^n
   +c_0B\sum_{r=0}^\infty
       2^{jn}2^{-rM}(2^{r-j+2})^n
 =c_0B\left(2^n+\frac{2^{2n}}{1-2^{n-M}}\right)<\infty .
\]
No rescaled unit cube or suppressed Jacobian is needed for this estimate.

For coordinate \(\ell\), differentiation of (MP5) gives the full factor \(i2^{j(n+1)}\) and the integrand \(\eta_\ell a_j(\eta)\). Its \(M\)-th derivative in coordinate \(i\) is
\(\eta_\ell\partial_i^Ma_j+M\delta_{i\ell}\partial_i^{M-1}a_j\).
Thus, with every product-rule term retained,
\[
 \begin{split}
 c_\ell&=(2\pi)^{-n}
   \max\{2P_0,\ n^{M/2}\max_i(2P_{i,M}
                           +M\delta_{i\ell}P_{i,M-1})\},\\
 |\partial_\ell K_j(x)|&\leq c_\ell B\,2^{j(n+1)}
                     \min\{1,(2^j|x|)^{-M}\}.
 \end{split}                                                   \tag{MP8}
\]
This is the original gradient scale \(2^{j(n+1)}\); no \(2\pi\) is inserted into the gradient under the phase \(e^{ix\cdot\xi}\). The inverse factor \((2\pi)^{-n}\) remains in each actual integral and cutoff constant.

For \(x\ne0\), choose the integer \(J\) with \(1/2<\rho=2^J|x|\leq1\). The low and high sums in (MP7) give, respectively,
\[
 \begin{split}
 |K(x)|&\leq c_0B|x|^{-n}
 \left(\frac{\rho^n}{1-2^{-n}}
            +\frac{(2\rho)^{-2}}{1-2^{-2}}\right)
       \leq A_0B|x|^{-n},\\
 |\nabla K(x)|&\leq
 B|x|^{-n-1}\left(\sum_{\ell=1}^nc_\ell^2\right)^{1/2}
 \left(\frac{\rho^{n+1}}{1-2^{-n-1}}
            +\frac{(2\rho)^{-1}}{1-2^{-1}}\right)
       \leq A_1B|x|^{-n-1},\\
 A_0&=c_0\left(\frac1{1-2^{-n}}+\frac1{1-2^{-2}}\right),\\
 A_1&=\left(\sum_{\ell=1}^nc_\ell^2\right)^{1/2}
       \left(\frac1{1-2^{-n-1}}+\frac1{1-2^{-1}}\right).
 \end{split}                                                   \tag{MP9}
\]
Here \(K=\sum_{j\in\mathbb Z}K_j\). Both series and the derivative series converge uniformly on every compact set avoiding zero, by the same positive geometric bounds; the finite fundamental theorem of calculus then proves that \(K\) is \(C^1\) there with the displayed derivative sum. The complete \(\rho\) contributions are retained before their uniform comparison.

### 13.3. Identifying the original operator outside the support

The full finite telescoping identity is
\[
 \sum_{j=-N}^Nb_j(\xi)
 =b(\xi)\bigl(\chi(2^{-N-1}\xi)-\chi(2^N\xi)\bigr)
 \longrightarrow b(\xi)\quad(\xi\ne0).
                                                               \tag{MP10}
\]
Its modulus is at most \(2\|\chi\|_\infty B\). Dominated convergence in the exact \(L^2\) Fourier pairing proves convergence of these finite multiplier operators to \(T\) in \(L^2\).

The full original distribution kernel is \(\mathcal K=Gb\), not a substitution of the off-diagonal function \(K\) at its singular point. For every Schwartz test \(\phi\), its exact complex-linear pairing is
\[
 \langle\mathcal K,\phi\rangle
   =(2\pi)^{-n}\int b(\xi)F\phi(-\xi)\,d\xi,\qquad
 \sum_{j=-N}^NK_j\longrightarrow\mathcal K
                         \quad\hbox{in }\mathcal S' .
                                                               \tag{MP10a}
\]
The integral is absolutely convergent because \(b\) is bounded and \(F\phi\) is Schwartz. The telescoping bound and dominated convergence prove the limit on every test. It also converges in the strong dual topology, uniformly on every bounded subset \(E\) of the original Schwartz space. Indeed, put \(H_L(E)=\sup_{\phi\in E,\xi}(1+|\xi|)^L|F\phi(\xi)|<\infty\), for an integer \(L>n\). The finiteness follows from the continuous original Fourier map and the finite expansion of the weight into its monomial seminorms. The difference of the actual finite symbol and \(b\) is
\(b[\chi(2^{-N-1}\xi)-1-\chi(2^N\xi)]\).
The last cutoff is supported in \(|\xi|\leq2^{-N}\), and the preceding cutoff difference vanishes for \(|\xi|\leq2^N\). Consequently,
\[
 \begin{split}
 \sup_{\phi\in E}\left|
    \left\langle\sum_{j=-N}^NK_j-\mathcal K,\phi\right\rangle\right|
 &\leq(2\pi)^{-n}B H_L(E)
 \left[\|\chi\|_\infty(2^{1-N})^n\right.\\
 &\left.\qquad+(1+\|\chi\|_\infty)
                \int_{|\xi|\geq2^N}(1+|\xi|)^{-L}\,d\xi\right]
 \longrightarrow0 .
 \end{split}                                                   \tag{MP10b}
\]
The low term uses its original containing cube; the high term tends to zero by absolute integrability of the displayed weight, proved by the original dyadic cube comparison. This verifies the precise distribution topology as well as the limit on individual tests. For a compact smooth test supported away from zero, the uniform kernel convergence already proved identifies this pairing with \(\int K\phi\). Hence the restriction of the actual full kernel to the complement of zero is exactly the regular distribution defined by \(K\). Its possible contribution supported at zero is retained by (MP10a).

The exact convolution map is also retained. For Schwartz \(f,\phi\), put \(\Theta(y)=\int f(x-y)\phi(x)\,dx\), whose full Schwartz bounds are proved in Section 13.10, GC19–GC24, of [Detecting regularity without choosing coordinates](geometric-microlocal-calculus.md). Absolute product integration gives \(F\Theta(-\xi)=Ff(\xi)F\phi(-\xi)\). Thus the full distributional convolution has the actual pairing
\[
 \langle\mathcal K*f,\phi\rangle
   =(2\pi)^{-n}\int b(\xi)Ff(\xi)F\phi(-\xi)\,d\xi
   =\langle G(bFf),\phi\rangle .
\]
This proves \(\mathcal K*f=Tf\) with its entire distribution kernel. For example, when \(b(\xi)=c_{\mathrm{const}}\) is constant, Fourier inversion in this same pairing gives \(\mathcal K=c_{\mathrm{const}}\delta_0\) and \(T=c_{\mathrm{const}}I\). The off-diagonal restriction is then zero, while the original operator remains present in the \(L^2\) and level-set calculation. No diagonal contribution is dropped.

For \(f\in L^2\) with compact support, Hölder gives \(f\in L^1\). Each finite term has the identity
\[
 G(b_jFf)(x)=\int K_j(x-y)f(y)\,dy .
                                                               \tag{MP11}
\]
To verify it, substitute (MP5) before its frequency rescaling, and apply absolute Fubini; the bound is \((2\pi)^{-n}\|b_j\|_1\|f\|_1\). The result is the ordinary inverse integral, equal to its \(L^2\) class by Section13.1 and [the Fourier chapter, Section7.2](prerequisite-bridges.md#AN03-DEP-L2-002). On a compact set separated from the input support, the series of these convolution integrals converges uniformly and absolutely, because the positive kernel sum is bounded there by \(A_0B\) times the fixed negative distance power and \(\|f\|_1\). A subsequence of the \(L^2\) operator convergence identifies the resulting function with \(Tf\) almost everywhere there. Exhaust the complement of the compact support by countably many separated compact sets. This proves
\(Tf(x)=\int K(x-y)f(y)\,dy\)
almost everywhere outside the support, for the original operator rather than a new kernel model.

### 13.4. Dyadic differentiation and decomposition proved from the entry measure

Keep the actual half-open dyadic cubes
\(2^{-j}(k+[0,1)^n)\), \(j\in\mathbb Z\), \(k\in\mathbb Z^n\).
Two such cubes are disjoint or one contains the other; this follows coordinatewise from the integer dyadic grid. For \(f\in L^1\), define its dyadic maximal function as the supremum of the absolute averages on all cubes containing the point. For each positive height \(a\), cubes whose average of \(|f|\) exceeds \(a\) have maximal members: their ancestors' volumes tend to infinity while their input integrals are at most \(\|f\|_1\), so sufficiently coarse ancestors have average at most \(a\). The maximal cubes are disjoint, form a countable family, and cover every high cube. Their exact original volumes give
\[
 |\{\mathcal M_df>a\}|\leq a^{-1}\|f\|_1 .
                                                               \tag{MP12}
\]

Let \(A_jf(x)\) be the average over its unique dyadic cube of side \(2^{-j}\). For compact continuous \(v\),
\(|A_jv(x)-v(x)|\leq\omega_v(\sqrt n\,2^{-j})\to0\).
The modulus tends to zero by uniform continuity on a large compact containing its support; outside that compact the same continuity follows from zero extension. For \(f\in L^1\), choose compact smooth \(v\) with arbitrarily small \(\|f-v\|_1\), by its density proof. The exact difference estimate is
\[
 \limsup_{j\to\infty}|A_jf(x)-f(x)|
       \leq\mathcal M_d(f-v)(x)+|f(x)-v(x)|.
                                                               \tag{MP13}
\]
For \(\delta>0\), (MP12) and the integral bound for \(|f-v|\) show that the set where the left side exceeds \(2\delta\) has measure at most \(2\|f-v\|_1/\delta\). Taking the infimum over these approximations gives zero. A countable union over positive rational \(\delta\) proves \(A_jf\to f\) almost everywhere. Apply the same proof to \(|f|\). This establishes the needed differentiation theorem without importing one.

For \(f\in L^1\cap L^2\) and a height \(a>0\), select its maximal high cubes \(Q\). Each satisfies
\[
 a<|Q|^{-1}\int_Q|f|\leq2^na,
 \qquad \sum_Q|Q|\leq a^{-1}\|f\|_1 .
                                                               \tag{MP14}
\]
The upper bound follows from its parent, whose original volume is \(2^n|Q|\) and whose average is at most \(a\). Write \(m_Q=|Q|^{-1}\int_Qf\), set \(g=f\) outside the union and \(g=m_Q\) on each cube, and put \(d_Q=(f-m_Q)1_Q\). Dyadic differentiation gives \(|f|\leq a\) almost everywhere outside the union. The original complex averages and disjoint cubes then give
\[
 \begin{split}
 f&=g+\sum_Qd_Q,\qquad \int d_Q=0,\qquad
 \|g\|_\infty\leq2^na,\qquad \|g\|_1\leq\|f\|_1,\\
 \|g\|_2^2&\leq2^na\|f\|_1,\qquad
 \sum_Q\|d_Q\|_1\leq2\|f\|_1 .
 \end{split}                                                   \tag{MP15}
\]
The sums converge in \(L^1\); they also converge in \(L^2\), because \(f-g\in L^2\) and its pieces have disjoint supports. In particular every \(d_Q\) is an actual compactly supported \(L^1\cap L^2\) function, with its original mean exactly zero.

### 13.5. Complete weak estimate with the actual symbol bound

If \(B=0\), (MP1) makes \(b=0\) away from its null exceptional point and \(T=0\). Otherwise fix the output level \(\lambda>0\) and use the decomposition at the actual input height \(a=\lambda/B\). This retains the original \(f\), \(b\), \(B\) and output level; none is replaced by a unit-norm object.

For a selected cube \(Q\) of side \(\ell_Q\) and center \(c_Q\), set
\(R_Q=2\sqrt n\,\ell_Q\) and
\(Q^*=c_Q+(-R_Q,R_Q)^n\).
Its original volume is \((4\sqrt n)^n|Q|\). If \(x\notin Q^*\) and \(y\in Q\), then
\(|x-c_Q|\geq R_Q\) and
\(|y-c_Q|\leq\sqrt n\,\ell_Q/2\leq|x-c_Q|/4\).
Every point of the segment between \(x-y\) and \(x-c_Q\) consequently has length at least \(3|x-c_Q|/4\).
The zero mean and (MP9) give
\[
 |Td_Q(x)|\leq A_1B(4/3)^{n+1}|x-c_Q|^{-n-1}
                        \int_Q|y-c_Q||d_Q(y)|\,dy .
                                                               \tag{MP16}
\]
The original kernel representation in Section13.3, established from (MP11), applies to each \(d_Q\).
On dyadic Euclidean shells beyond \(R_Q\), use the original containing coordinate cubes to get
\[
 \int_{|x-c_Q|\geq R_Q}|x-c_Q|^{-n-1}\,dx
 \leq\sum_{j=0}^\infty
            (2^jR_Q)^{-n-1}(2^{j+2}R_Q)^n
 =\frac{2^{2n+1}}{R_Q}.
                                                               \tag{MP17}
\]
Sphere boundaries are null by the coordinate-section proof in FI8. Therefore
\[
 \int_{(Q^*)^c}|Td_Q|
 \leq A_1B(4/3)^{n+1}
        \frac{\sqrt n\,\ell_Q}{2}\frac{2^{2n+1}}{R_Q}\|d_Q\|_1
 =D_nB\|d_Q\|_1,\qquad
 D_n=2^{2n-1}(4/3)^{n+1}A_1 .
                                                               \tag{MP18}
\]
Both actual radius factors and all derivative constants remain explicit.

Let \(\Omega^*=\bigcup_QQ^*\). The series of \(Td_Q\) converges absolutely in \(L^1((\Omega^*)^c)\) by (MP15)–(MP18). Its \(L^2\) partial sums also converge to \(T(f-g)\). Almost-everywhere subsequences identify the two limits, so
\(\int_{(\Omega^*)^c}|T(f-g)|\leq2D_nB\|f\|_1\).
The \(L^2\) estimate of \(Tg\), the enlarged volume sum and the last integral estimate now give
\[
 \begin{split}
 |\{|Tf|>\lambda\}|
 &\leq|\Omega^*|+|\{|Tg|>\lambda/2\}|
       +|\{x\notin\Omega^*:|T(f-g)|>\lambda/2\}|\\
 &\leq\left((4\sqrt n)^n+4\,2^n+4D_n\right)
                   \frac{B}{\lambda}\|f\|_1
   =W_n\frac{B}{\lambda}\|f\|_1 .
 \end{split}                                                   \tag{MP19}
\]
This weak estimate is proved for every \(f\in L^1\cap L^2\), precisely the domain needed in the next argument.

### 13.6. Strong finite exponents without an interpolation import

Let \(1<p<2\), initially \(f\in L^p\cap L^2\). At each \(\lambda>0\), split it at its actual value threshold \(\lambda/B\) into \(f_{\rm high}\) and \(f_{\rm low}\).
The high part is in \(L^1\cap L^2\), because
\(|f|1_{\{|f|>\lambda/B\}}\leq(\lambda/B)^{1-p}|f|^p\).
The low part is in \(L^2\), including for an arbitrary \(L^p\) input, by
\(|f_{\rm low}|^2\leq(\lambda/B)^{2-p}|f|^p\).
Linearity in \(L^2\), (MP19) and (MP3) give
\[
 |\{|Tf|>\lambda\}|
 \leq\frac{2W_nB}{\lambda}
             \int_{\{|f|>\lambda/B\}}|f|
       +\frac{4B^2}{\lambda^2}
             \int_{\{|f|\leq\lambda/B\}}|f|^2 .
                                                               \tag{MP20}
\]
Use the level-set identity proved in FI5. Nonnegative Fubini retains every threshold factor:
\[
 \begin{split}
 \|Tf\|_p^p
 &\leq2W_nBp\int|f(x)|
                    \int_0^{B|f(x)|}\lambda^{p-2}\,d\lambda\,dx\\
 &\quad+4B^2p\int_{\{f\ne0\}}|f(x)|^2
                    \int_{B|f(x)|}^{\infty}\lambda^{p-3}\,d\lambda\,dx\\
 &=\left(\frac{2W_np}{p-1}+\frac{4p}{2-p}\right)
                         B^p\|f\|_p^p .
 \end{split}                                                   \tag{MP21}
\]
At \(f=0\), the second contribution is zero, so no product of zero with an infinite scalar integral is used. The conditions \(p>1\) and \(p<2\) are exactly the two scalar endpoint integrability conditions. Define
\(L_{n,p}=(2W_np/(p-1)+4p/(2-p))^{1/p}\).
On Schwartz inputs this proves the original norm bound. Density and complete \(L^p\) extend \(T\) uniquely to all \(L^p\), with norm at most \(L_{n,p}B\). Its \(L^p\) and \(L^2\) realizations agree on the intersection: use simultaneous smooth approximation in both norms and identify their limits in measure, or by an almost-everywhere subsequence. At \(p=2\), use the exact bound \(B\) in (MP3).

For \(p>2\), let \(r=p/(p-1)\in(1,2)\). The original conjugate symbol satisfies the same complete derivative bounds (MP1), since real-coordinate derivatives commute with complex conjugation. Hence the just-proved theorem applies to \(T^*\) with norm at most \(L_{n,r}B\) on \(L^r\), and it agrees with its \(L^2\) realization on bounded compactly supported inputs by simultaneous mollification, as proved in Section 7.2 of the Fourier chapter.

For Schwartz \(f\), put \(u=Tf\in L^2\). On
\(E_{R,N}=\{|x|_\infty\leq R,\ |u(x)|\leq N\}\), define the actual test
\(h_{R,N}=1_{E_{R,N}}|u|^{p-2}u\), setting it to zero at \(u=0\).
It is bounded and supported in a finite coordinate cube, so belongs to \(L^2\cap L^r\). The exact \(L^2\) adjoint identity and Hölder give
\[
 \int_{E_{R,N}}|u|^p
   =\langle u,h_{R,N}\rangle
   =\langle f,T^*h_{R,N}\rangle
   \leq L_{n,r}B\|f\|_p
                \left(\int_{E_{R,N}}|u|^p\right)^{1/r}.
                                                               \tag{MP22}
\]
Divide only if the finite integral is positive; the zero case is direct. Increasing \(R,N\) to infinity and applying the proved monotone-convergence theorem proves
\(\|Tf\|_p\leq L_{n,r}B\|f\|_p\).
Density and completeness again extend this original operator to the entire \(L^p\) space. This uses its actual nonlinear test function, rather than assuming an \(L^p\) dual representation.

Thus the original multiplier theorem is fully proved with
\[
 \|b(D)f\|_p\leq C^{\rm mult}_{n,p,\chi}B\|f\|_p,\qquad
 C^{\rm mult}_{n,p,\chi}=
 \begin{cases}
 L_{n,p},&1<p<2,\\
 1,&p=2,\\
 L_{n,p/(p-1)},&2<p<\infty .
 \end{cases}                                                   \tag{MP23}
\]
The displayed cutoff dependence records the actual chosen cutoff. Fixing one original cutoff for each dimension makes it a dimension/exponent constant in the original statement; all working formulas retain that cutoff and its derivative norms.

### 13.7. The proved derivative-bound underclaim

The proof used, among the given frequency derivative bounds, only
\[
 B_{\rm coord}=\max_{\substack{1\leq i\leq n\\0\leq r\leq n+2}}
         \sup_{\xi\ne0}|\xi|^r|\partial_i^rb(\xi)| .
                                                               \tag{MP24}
\]
For a symbol still in \(C^{n+2}(\mathbb R^n\setminus\{0\})\), the finiteness of this smaller displayed maximum suffices, even if no bound is imposed on its mixed derivatives. Each Leibniz and integration-by-parts estimate above uses one original coordinate repeatedly; the output-gradient identity adds only its explicitly displayed zeroth and first coordinate-monomial terms. Every later step uses just the resulting original kernel/gradient bounds and the zeroth symbol bound. Replacing the bound parameter in that exact proof by \(B_{\rm coord}\) therefore proves the stronger statement with the same cutoff constants. The original full \(B\) and every one of its hypotheses remain visible in the comparison, and \(B_{\rm coord}\leq B\) recovers (MP23). This is a proved consequence of the calculation; it is presented here as an explicit addition while the original statement remains identifiable.

### 13.8. Full constants in the original elliptic estimates

For the original frozen polynomial \(q\) in (L1), retain \(b_0=(1-\chi)/q\) and every constant \(C_\nu\) in (L3). At top order \(|\alpha|=m\), its symbol is \(b_\alpha(\xi)=\xi^\alpha b_0(\xi)\). For every \(|\beta|\leq n+2\), the exact finite Leibniz formula and its weighted estimate are
\[
 \begin{split}
 \partial^\beta b_\alpha(\xi)
 &=\sum_{\substack{\gamma\leq\beta\\\gamma\leq\alpha}}
       \binom{\beta}{\gamma}\frac{\alpha!}{(\alpha-\gamma)!}
       \xi^{\alpha-\gamma}\partial^{\beta-\gamma}b_0(\xi),\\
 |\xi|^{|\beta|}|\partial^\beta b_\alpha(\xi)|
 &\leq\sum_{\substack{\gamma\leq\beta\\\gamma\leq\alpha}}
       \binom{\beta}{\gamma}\frac{\alpha!}{(\alpha-\gamma)!}
       C_{\beta-\gamma}
       \frac{|\xi|^{m+|\beta|-|\gamma|}}
            {(1+|\xi|)^{m+|\beta|-|\gamma|}}\\
 &\leq B_\alpha,\qquad
 B_\alpha=\max_{|\beta|\leq n+2}
       \sum_{\substack{\gamma\leq\beta\\\gamma\leq\alpha}}
       \binom{\beta}{\gamma}\frac{\alpha!}{(\alpha-\gamma)!}
       C_{\beta-\gamma}.
 \end{split}                                                   \tag{MP25}
\]
The exponent in every ratio is nonnegative, since
\(|\gamma|\leq|\beta|\); no polynomial factor or reciprocal derivative is discarded. The symbol is smooth at zero because the numerator of \(b_0\) vanishes on its original neighborhood. Thus (MP23) gives
\[
 \begin{split}
 \|D^\alpha F_0*g\|_p
 &\leq C^{\rm mult}_{n,p,\chi}B_\alpha\|g\|_p,\\
 \left\|\left(\sum_{|\alpha|=m}
               |D^\alpha F_0*g|^p\right)^{1/p}\right\|_p
 &\leq C^{\rm mult}_{n,p,\chi}
            \left(\sum_{|\alpha|=m}B_\alpha^p\right)^{1/p}\|g\|_p .
 \end{split}                                                   \tag{MP26}
\]
There are exactly \(\binom{n+m-1}{m}\) original multiindices in this array. Its equality with the sum of the \(p\)-th powers of the component norms follows by integrating the finite sum; no component is suppressed. On a compactly supported input the distributional convolution agrees with the multiplier. To prove this for the actual \(L^p\) input, mollify its zero extension with uniformly bounded compact support. The sequence converges in \(L^p\), its multipliers converge in \(L^p\), and convolution with the original tempered distribution converges distributionally by the compact-factor test-function bounds in the convolution chapter. Differentiation and uniqueness of the distributional limit identify the two operators.

For lower orders \(|\alpha|<m\), put \(k=m-|\alpha|>0\). The exact annular scaling in (L6) gives
\(\|K_\ell\|_1=2^{-\ell k}\|\kappa_\alpha\|_1\).
Define the actual operator constants
\[
 J_\alpha=
 \begin{cases}
 C^{\rm mult}_{n,p,\chi}B_\alpha,&|\alpha|=m,\\
 \displaystyle\frac{\|\kappa_\alpha\|_1}{1-2^{-(m-|\alpha|)}},
                                      &|\alpha|<m,
 \end{cases}
 \quad
 \Omega_\alpha=\|D^\alpha\omega\|_1,\quad
 C_q^{\rm est}=
 \max\left\{\sum_{|\alpha|\leq m}J_\alpha,\
            \sum_{|\alpha|\leq m}\Omega_\alpha\right\}.
                                                               \tag{MP27}
\]
The inverse Fourier factor is retained in the original
\(\kappa_\alpha=\mathcal F^{-1}(\psi\xi^\alpha/q)\) and
\(\omega=-\mathcal F^{-1}\chi\); their displayed norms are not norms of substituted kernels. Integrating each original (L14) by the proved Young inequality and (MP26) gives its estimate with \(C_q^{\rm est}\) at \(A=1\).
For every original \(A>0\), the actual comparison
\(w(y)=v(y/A)\) gives
\(\|D^\alpha w\|_p=A^{n/p-|\alpha|}\|D^\alpha v\|_p\),
\(\|q(D)w\|_p=A^{n/p-m}\|q(D)v\|_p\), and
\(\|w\|_p=A^{n/p}\|v\|_p\).
Multiply the obtained inequality by \(A^{m-n/p}\). This proves exactly (L13), with every displayed \(A^{m-|\alpha|}\) and \(A^m\) term, using \(C_q^{\rm est}\), for its whole stated \(W^{m,p}\) domain by the already proved approximation.

The constants can be uniform on an original compact family of elliptic \(q\)'s. The function \((q,\theta)\mapsto|q(\theta)|\), on that family times the unit sphere, has a positive minimum by compactness and ellipticity; the finitely many polynomial coefficients have a finite maximum. All derivatives of \(q\) on the fixed annulus are therefore uniformly bounded. The exact reciprocal recurrence
\[
 \partial^\nu(1/q)
 =-q^{-1}\sum_{\substack{0<\gamma\leq\nu}}
       \binom{\nu}{\gamma}
       (\partial^\gamma q)\,
       \partial^{\nu-\gamma}(1/q),\qquad |\nu|>0
                                                               \tag{MP28}
\]
retains every product and its negative sign. Induction gives uniform bounds for each derivative needed in (L3), with homogeneity transporting the original powers from the sphere and the fixed cutoff region treated by the same recurrence. It also uniformly bounds every derivative of the compact annular symbol defining \(\kappa_\alpha\). For an integer \(N>n\), coordinate integration by parts as in (MP7) bounds that kernel by its original direct integral near zero and a constant times \(|x|^{-N}\) outside; the full containing-cube sums integrate this bound. Thus each \(\|\kappa_\alpha\|_1\), \(B_\alpha\) and \(C_q^{\rm est}\) is uniform. The fixed \(\omega\) is unchanged.

Finally retain the smooth correction \(H=G-F_0\) in the local solver. For \(V\subset B_1\), the complete top-order array satisfies
\[
 \|\nabla^m S_Vg\|_{L^p(V)}
 \leq\left[
 C^{\rm mult}_{n,p,\chi}
       \left(\sum_{|\alpha|=m}B_\alpha^p\right)^{1/p}
 +\left(\sum_{|\alpha|=m}
                 \sup_{B_2}|D^\alpha H|^p\right)^{1/p}
       |V|^{1/p}|B_1|^{1-1/p}
 \right]\|g\|_{L^p(V)} .
                                                               \tag{MP29}
\]
Every difference \(x-y\) lies in the original \(B_2\); Hölder bounds its input integral by the original \(|B_1|^{1-1/p}\), and integrating its constant output over \(V\) gives \(|V|^{1/p}\). This is the exact \(k=0,q=p\) case of (L11) and the missing top-order part of (L17). The original perturbative inverse \(B_V\) there has norm at most \(2\). Substituting \(g=B_Vf\) in (MP29) consequently bounds the same array of \(Ef=S_VB_Vf\) by twice its displayed bracket times \(\|f\|_{L^p(V)}\). Lower orders retain their proved fractional or strict Young estimates and the original smooth corrections in Section 12.5. Both inverse identities, each coefficient product and all original distance weights remain those in Sections3–9.

## References

The mathematical antecedent for the local coefficient theorem and the subordinate estimates is Lars Hörmander, *The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators*, corrected second printing, 1994, §17.1. In particular, the finite-part construction, the scaled Hölder proof and the explicit annular cover expose steps needed for the stated generality. The earlier Vol. II antecedent has its separate interface in constant-strength antecedent.

Tao's [*Lecture Notes 4 for 247A*](https://www.math.ucla.edu/~tao/247a.1.06f/notes4.pdf), Theorem 4.4 and Corollary 2.10, is the specified multiplier reading. His [*Lecture Notes 2 for 247A*](https://www.math.ucla.edu/~tao/247a.1.06f/notes2.pdf), Proposition 6.1 and Corollary 6.3, is the specified fractional-integration reading. Their normalization uses \(2\pi\) in the Fourier phase; the change of frequency variable gives our convention without changing the multiplier or exponent conditions. The cited multiplier and fractional-integration statements are used with the hypotheses stated in Section 1.

In the multiplier proof on page 20 of Notes 4, read the two intermediate gradient formulas with the frequency and scale factors restored:
\[
\nabla K_j=(2\pi i\xi m_j)^\vee,\qquad
|\nabla K_j(x)|\lesssim 2^{j(d+1)}
       \min\bigl(1,(2^j|x|)^{-(d+2)}\bigr).
\]
Here \(d\) is that reading's dimension. Its subsequent expression with the factor \(|x|^{-d-1}\) already has the correct scaling. These printing corrections preserve the theorem and its \(d+2\)-derivative hypothesis.

For a complementary proof technique, John K. Hunter's [*Notes on Partial Differential Equations*, revised 18 June 2014](https://www.math.ucdavis.edu/~hunter/pdes/pde_notes.pdf), Theorem 2.28 estimates the Newtonian potential in Hölder spaces by splitting spatial integrals. Its scope is the Laplacian; (L24)–(L26) handle arbitrary homogeneous elliptic order. Hunter's Lemma 8.11 treats a commutator for first-order systems with \(C^1\) coefficients in \(L^2\). Our proof in (L38)–(L39) gives the scalar Lipschitz, arbitrary finite \(p\) version used here. The comparisons do not substitute the smoother assumptions of those examples for the hypotheses of this lesson.

The earlier external multiplier and fractional-integration routes remain identifiable in the citations. Section 12 supplies an owned fractional-integration proof with its original kernel, finite endpoint, full constants and exact elliptic application. Section 13 supplies the full multiplier proof, including the complete Fourier maps and every top-order elliptic receiving constant. The general constant-strength antecedent retains its separate dependency on the Vol. II constant-coefficient theory.