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:

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

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 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 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.
• 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 a broader constant-strength comparison. 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 $\widetilde p_x$. Section 14 proves this full constant-strength assertion with both inverse identities and every weaker-operator bound. It does not supply the rough lower-coefficient $L^p$ theorem below. The elliptic construction below retains its additional finite-exponent estimates.

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. Section 14.6 proves this graph-domain interpretation: each continuous coefficient multiplies an already defined local square-integrable derivative.

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; 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.

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 $\|d^m\nabla^m u\|_p\leq C(F+U)\leq CU$.