Regular kernels and changes in the equation

Written by GPT-6.1 Sol (OpenAI), Ultra reasoning effort, October 2026. Self-checked by GPT-6.1 Sol (OpenAI). Public domain (CC0).

Solving an equation is only part of the problem. We also want the solution to retain the regularity of the data, and we want estimates that survive a change in the coefficients. These two questions meet in the choice of a fundamental solution. A poorly chosen inverse can carry avoidable singularities. A regular inverse uses the full derivative norm of the polynomial to control its localized Fourier transform.

Read Measuring regularity with weighted Fourier spaces and Local regularity, sharp embeddings, and compactness first. We also use the distributional convolution identities discussed in Fundamental solutions and the directions an equation uses. The additional prerequisite is holomorphic averaging for polynomials, stated precisely below. Its construction belongs to the theory of Fourier–Laplace transforms. Basic references are the open treatments of distributions by Grubb [Grubb] and Melrose [Melrose], Malgrange's original existence paper [Malgrange], Hörmander's survey of existence and regularity [Hormander 1971], and Mantlik's work on dependence on a parameter [Mantlik].

We keep \(D=-i\partial\), the negative-exponential Fourier transform, and complex-linear distributional pairings. For a nonzero polynomial, \[ S_P(\xi)=\left(\sum_\alpha|\partial^\alpha P(\xi)|^2\right)^{1/2}. \] All polynomial spaces in this lesson have a fixed maximum degree \(m\). The actual degree may fall when the coefficients vary.

The smooth polynomial averaging statement below is proved in Averaging an entire function while avoiding polynomial zeros, including compact averaging support, scalar homogeneity and the uniform denominator bound. The construction retains its declared scalar Taylor, finite-dimensional compactness, cutoff and Lebesgue/Fubini inputs; the other prerequisites and recursive closure of this lesson remain explicit.

What a regular inverse controls

A fundamental solution \(E\) of \(P(D)\) is regular if \[ E\in B_{\infty,S_P}^{\mathrm{loc}}(\mathbb R^n). \] Thus every compactly supported smooth \(\chi\) gives a bounded function \(S_P\widehat{\chi E}\). This is a local assertion. It makes no claim that \(E\) is tempered before a cutoff is applied.

We shall use a simple Fourier estimate repeatedly. Suppose \(a\) is measurable and \(\sup_\xi k(\xi)|a(\xi)|\leq A\), and let \(g=\mathcal F^{-1}a\). For \(f\in\mathcal S\), \[ \widehat{fg}(\eta)=(2\pi)^{-n}\int\widehat f(h)a(\eta-h)\,dh, \] so \[ \|fg\|_{\infty,k}\leq A(2\pi)^{-n}\int M_k(h)|\widehat f(h)|\,dh. \tag{1} \] Indeed \(k(\eta)/k(\eta-h)\leq M_k(h)\). Polynomial growth of \(M_k\) makes the integral finite. The formula first follows by testing against Schwartz functions and using absolute convergence; it then represents the Fourier transform by a bounded weighted function. In particular it applies when \(g\) is a tempered distribution rather than an ordinary function.

The exponential denominator we use is \[ w_\varepsilon(x)=\cosh(\varepsilon|x|). \] It is smooth at the origin: its power series contains only powers of \(|x|^2\). For fixed \(0<\rho<\varepsilon\), the functions \[ f_z(x)=\frac{e^{ix\cdot z}}{w_\varepsilon(x)}, \qquad z\in\mathbb C^n,\quad |z|\leq\rho, \tag{2} \] form a bounded subset of \(\mathcal S\). Every derivative is a sum of bounded powers of \(z\) times derivatives of \(1/w_\varepsilon\). Away from the origin those derivatives decay exponentially, and the factor \(e^{ix\cdot z}\) grows by at most \(e^{\rho|x|}\). The remaining decay is \(e^{-(\varepsilon-\rho)|x|}\), which absorbs every polynomial in \(x\).

Holomorphic averaging as a starting tool

Here is the precise prerequisite. Let \(\mathcal P_m\) be the complex vector space of polynomials of degree at most \(m\), and give it the norm \[ |q|_J=\left(\sum_{|\alpha|\leq m}|\partial^\alpha q(0)|^2\right)^{1/2}. \] For every \(\rho>0\), there is a nonnegative smooth function \(\Phi(q,z)\), defined for \(q\ne0\) and \(z\in\mathbb C^n\), with these properties:

  1. Its \(z\)-support is contained in a fixed compact subset of \(\{|z|<\rho\}\).
  2. \(\Phi(aq,z)=\Phi(q,z)\) for every nonzero complex scalar \(a\).
  3. For every entire holomorphic \(H\), \[ \int_{\mathbb C^n}H(z)\Phi(q,z)\,d\lambda(z)=H(0). \tag{3} \]
  4. There is a constant \(c>0\) such that \[ |q(z)|\geq c|q|_J\quad\hbox{whenever }\Phi(q,z)\ne0. \tag{4} \]

Here \(\lambda\) is real \(2n\)-dimensional Lebesgue measure. The constants depend on \(m,n,\rho\). The complete polynomial averaging proof supplies this lemma, including its smooth dependence on the real and imaginary coefficient coordinates and the denominator differentials below. The averaging construction is due to Hörmander; a reference for the existence and regularity method is [Hormander 1971]. The existence theorem it supports is the Malgrange–Ehrenpreis theorem. The argument below derives the stronger estimates from the stated averaging properties.

Define \[ A(q,z)=\frac{\Phi(q,z)}{q(z)} \] where \(q(z)\ne0\), and define it to be zero near the zero set of \(q(z)\). This is a smooth definition. In fact (4) says that \(\Phi\) vanishes on the open set \(|q(z)|<c|q|_J\), including a neighborhood of every zero of \(q(z)\).

Lemma 2.1. For every \(j\geq0\), the real coefficient differential of \(A\) satisfies \[ \big|d^jA(q,z)[r_1,\ldots,r_j]\big| \leq C_j\frac{\prod_{\nu=1}^j|r_\nu|_J}{|q|_J^{j+1}}. \tag{5} \] The empty product is one. All these differentials have \(z\)-support in the same fixed compact set.

Proof. On the unit coefficient sphere \(|q|_J=1\), the function \(A\) and all of its coefficient differentials are bounded on the fixed compact \(z\)-set. Smoothness across the division set was just established. Outside that set they vanish. If \(t>0\), then \(A(tq,z)=t^{-1}A(q,z)\). Differentiating this identity \(j\) times scales the differential by \(t^{-j-1}\). Normalize \(q\) to the unit sphere and use the operator norm of the resulting multilinear differential. This proves (5). \(\square\)

For a translated polynomial \(P_\xi(z)=P(\xi+z)\), its coefficient norm is exactly \(S_P(\xi)\). Translation of the directions gives \(|(Q_\nu)_\xi|_J=S_{Q_\nu}(\xi)\). Therefore (5) becomes a frequency estimate, uniform over all nonzero \(P\): \[ \big|d^jA(P_\xi,z)[(Q_1)_\xi,\ldots,(Q_j)_\xi]\big| \leq C_j\frac{\prod_\nu S_{Q_\nu}(\xi)}{S_P(\xi)^{j+1}}. \tag{6} \] No lower bound on an individual coefficient of \(P\) is needed.

A smooth family with small exponential growth

Theorem 3.1. Fix \(m,n\) and \(\rho>0\), and choose one averaging function with support in \(|z|<\rho\). There is a real \(C^\infty\) map \[ P\longmapsto E_\rho(P),\qquad \mathcal P_m\setminus\{0\}\longrightarrow\mathcal D'(\mathbb R^n), \] such that \(P(D)E_\rho(P)=\delta_0\). For every \(\varepsilon>\rho\), \[ \left\|\frac{E_\rho(P)}{w_\varepsilon}\right\|_{\infty,S_P} \leq C_0. \tag{7} \] For nonzero \(Q_1,\ldots,Q_j\in\mathcal P_m\), put \[ F_j(\xi)=\frac{S_P(\xi)^{j+1}}{\prod_{\nu=1}^jS_{Q_\nu}(\xi)}. \] Then \[ \left\|\frac{E_\rho^{(j)}(P)[Q_1,\ldots,Q_j]}{w_\varepsilon}\right\|_{\infty,F_j} \leq C_j. \tag{8} \] The constants are independent of the coefficients of \(P,Q_1,\ldots,Q_j\). They depend on the fixed averaging choice, \(m,n,j,\varepsilon\). A zero direction makes the differential zero. Every \(E_\rho(P)\) is regular and has support in the active subspace \(V_P=N_P^\perp\).

Proof. Define the distribution by \[ \langle E_\rho(P),\phi\rangle =(2\pi)^{-n}\int_{\mathbb R^n}\int_{\mathbb C^n} \widehat\phi(-\xi-z)A(P_\xi,z)\,d\lambda(z)\,d\xi, \qquad\phi\in C_c^\infty. \tag{9} \] For a fixed \(P\ne0\), \(S_P\) is bounded below by a positive constant, because a derivative of its highest nonzero homogeneous part is a nonzero constant. The inner integrand is bounded by \(C/S_P(\xi)\), apart from the rapidly decreasing test transform. The transform \(\widehat\phi(-\xi-z)\) decreases faster than any power of \(|\xi|\), uniformly for \(z\) in the fixed compact set. This follows by integration by parts in \(x\), with its imaginary-frequency growth bounded on \(\operatorname{supp}\phi\). Thus (9) converges absolutely and defines a distribution.

The Fourier–Laplace transform of \(P(-D)\phi\), evaluated at \(-\xi-z\), is \(P(\xi+z)\widehat\phi(-\xi-z)\). Cancel that polynomial in (9) and apply (3) to the entire function \(z\mapsto\widehat\phi(-\xi-z)\). Fourier inversion gives \[ \langle P(D)E_\rho(P),\phi\rangle =(2\pi)^{-n}\int\widehat\phi(-\xi)\,d\xi =\phi(0). \]

For each \(z\), let \(g_z\) be the tempered distribution with transform \(A(P_\xi,z)\). Formula (9) can also be written \[ E_\rho(P)=\int e^{ix\cdot z}g_z\,d\lambda(z) \quad\hbox{in distributions}. \tag{10} \] After division by \(w_\varepsilon\), the multiplier in (10) is the Schwartz function (2). Apply (1) with \(k=S_P\), using (6) for \(j=0\). The shift function \(M_{S_P}\) has a polynomial bound with constants depending only on \(m,n\), by Proposition 1.3 of the weighted-spaces lesson. The functions \(f_z\) are uniformly Schwartz, and the \(z\)-set has finite measure. Integrating (1) proves (7), and also shows that the quotient in (7) is tempered. Multiplication by \(\chi w_\varepsilon\), for any \(\chi\in C_c^\infty\), proves local regularity.

To differentiate (9), observe that on a compact subset of \(\mathcal P_m\setminus\{0\}\), \[ S_P(\xi)\geq C^{-1}(1+|\xi|)^{-m}S_P(0) \geq c_K(1+|\xi|)^{-m}. \] The first inequality is the reverse moderate shift estimate. For bounded coefficient directions, \(S_Q(\xi)\) is bounded by a fixed polynomial in \(|\xi|\). Consequently every order of coefficient differentiation in (9) has a polynomially bounded integrand before multiplication by the test transform. The rapid decrease of that transform justifies differentiation under the integrals. The same estimates are uniform on bounded sets of test functions with a common compact support. This proves smoothness in the strong distribution topology, as well as the formula obtained by replacing \(A\) by its corresponding differential.

Now apply (1) to that differentiated formula. Estimate (6) bounds its Fourier coefficient by \(C_j/F_j\). Products and reciprocals of the weights \(S_Q\) are moderate. Their shift constants are uniform for degrees at most \(m\), so \[ M_{F_j}(h)\leq(1+C|h|)^{m(2j+1)}. \] Equations (1) and (2) therefore prove (8).

Finally, split the Euclidean coordinates as \(V_P\oplus N_P\). The polynomial \(P(\xi+z)\), and hence the translated polynomial \(P_\xi\) as an element of \(\mathcal P_m\), is independent of the \(N_P\)-component of \(\xi\). It follows that \(\widehat g_z\) is independent of that component. Its inverse transform is a distribution in the active coordinates tensored with \(\delta_0\) in the inactive coordinates. Thus \(g_z\), every multiplier in (10), and their integral are supported in \(V_P\). \(\square\)

For any prescribed positive exponential rate \(\varepsilon\), choose \(\rho<\varepsilon\). The theorem gives a family with the bound (7) at that rate. A fixed averaging choice has a fixed \(\rho\); the proof does not assert (7) for all smaller rates with that same choice. The parameter map is smooth in real coefficient coordinates. We have made no claim of holomorphic dependence on unrestricted complex coefficients.

This support statement strengthens the existence result in the first lesson: the family now has both minimal active-subspace support and regularity. The minimality theorem there says that no real linear subspace smaller than \(V_P\) can support a fundamental solution.

How differentiation forces the weight

The preceding estimate has a useful converse. It is independent of the averaging construction.

Proposition 4.1. Suppose \(P\mapsto E(P)\) is a real \(C^j\) family of fundamental solutions on a neighborhood of \(P\ne0\) in \(\mathcal P_m\). For coefficient directions \(Q_1,\ldots,Q_j\), \[ P(D)^{j+1}E^{(j)}(P)[Q_1,\ldots,Q_j] =(-1)^j j!\,Q_1(D)\cdots Q_j(D)\delta_0. \tag{11} \]

Proof. The case \(j=0\) is the defining equation. Assume the formula at order \(j-1\), with the first \(j-1\) directions. Differentiate it in direction \(Q_j\). Its right side is independent of \(P\), while its left side gives \[ jP(D)^{j-1}Q_j(D)E^{(j-1)} +P(D)^jE^{(j)}=0. \] Multiply by \(P(D)\) and substitute the induction formula for \(P(D)^jE^{(j-1)}\). Commutation of all constant-coefficient operators yields (11). \(\square\)

The regularity comparison in the next theorem uses the following consequence of convolution. If \(Q\ne0\), \(G\) is a regular fundamental solution of \(Q(D)\), and \(h\) is compactly supported with \(h\in B_{p,l}\), then \[ G*h\in B_{p,lS_Q}^{\mathrm{loc}}. \tag{12} \] For a cutoff \(\chi\), choose a compactly supported cutoff \(\psi\) equal to one near \(\operatorname{supp}\chi-\operatorname{supp}h\). On \(\operatorname{supp}\chi\), \(G*h=(\psi G)*h\). The compact kernel \(\psi G\) lies in \(B_{\infty,S_Q}\). The weighted convolution estimate and then the cutoff estimate prove (12), including \(p=\infty\).

Theorem 4.2. Assume every direction \(Q_\nu\) is nonzero. If the differential in Proposition 4.1 belongs to \(B_{p,k}^{\mathrm{loc}}\), for a moderate weight \(k\) and \(1\leq p\leq\infty\), then \[ \frac{k}{F_j}\in L^p(\mathbb R^n), \qquad B_{\infty,F_j}\subset B_{p,k} \quad\hbox{continuously}. \tag{13} \] For \(j=0\), this says that every local \(B_{p,k}\) regularity assertion for a fundamental solution forces \(k/S_P\in L^p\).

Proof. The differential-operator estimate sends the left side of (11) into \(B_{p,k/S_P^{j+1}}^{\mathrm{loc}}\). Therefore the compact distribution \(h_0=Q_1(D)\cdots Q_j(D)\delta_0\) belongs to that local space. Local and global membership agree for a distribution of fixed compact support.

Choose a regular fundamental solution \(G_\nu\) for each \(Q_\nu(D)\). Distributional convolution gives \[ G_1*h_0=Q_2(D)\cdots Q_j(D)\delta_0. \] Although \(G_1\) need not have compact support, this convolution is defined because \(h_0\) does; the displayed identity shows that its result does have compact support. Equation (12) improves its weight by \(S_{Q_1}\). Repeat the argument for \(G_2,\ldots,G_j\), converting local to global membership at each compact result. We obtain \[ \delta_0\in B_{p,k\prod_\nu S_{Q_\nu}/S_P^{j+1}} =B_{p,k/F_j}. \] Since \(\widehat\delta_0=1\), this is exactly the first assertion of (13). If \(u\in B_{\infty,F_j}\), then \[ \|u\|_{p,k} \leq(2\pi)^{-n/p}\|k/F_j\|_{L^p}\|u\|_{\infty,F_j}, \] with the factor interpreted as one for \(p=\infty\). This proves the inclusion. \(\square\)

The conclusion concerns weighted Fourier spaces. It does not assert that all fundamental solutions have the same wave front set or the same behavior at infinity. It also explains why the derivative norm \(S_Q\), rather than just \(|Q|\), appears in (8): the proof must invert the full operator \(Q(D)\), even at frequencies where its symbol vanishes.

A transport model with a visible regularity estimate

Let \(v\in\mathbb R^n\setminus\{0\}\), \(c\in\mathbb C\), and \[ L=v\cdot\nabla+c=P(D),\qquad P(\xi)=iv\cdot\xi+c. \] The forward kernel from the first lesson is \[ \langle E_+,\phi\rangle=\int_0^\infty e^{-ct}\phi(tv)\,dt. \] Its support is the ray \(\{tv:t\geq0\}\), and \(LE_+=\delta_0\). This definition is valid for every \(c\), because test functions have compact support along the ray.

For \(\chi\in C_c^\infty\), put \(h(t)=e^{-ct}\chi(tv)\). Then \[ \widehat{\chi E_+}(\xi)=\int_0^\infty h(t)e^{-itv\cdot\xi}\,dt. \] The \(L^1\) norm of \(h\) bounds this integral. For \(a=v\cdot\xi\ne0\), integration by parts also gives \[ \left|\int_0^\infty h(t)e^{-iat}\,dt\right| \leq\frac{|h(0)|+\|h'\|_{L^1(0,\infty)}}{|a|}. \] Combining the two estimates yields a bound by \(C/(1+|a|)\). Since \[ S_P(\xi)^2=|iv\cdot\xi+c|^2+|v|^2, \] the weight \(S_P\) is comparable to \(1+|v\cdot\xi|\). Hence \(E_+\) is regular. Its transform after a cutoff need not decay in the inactive frequency directions. The anisotropic weight records exactly that fact.

Exercises with solutions

Exercise 1 (entry). For a nonzero constant \(P=a\), identify the fundamental solution, the regularity weight, and the differentials in constant directions \(Q_\nu=b_\nu\).

Solution. The equation has the unique fundamental solution \(E(a)=a^{-1}\delta_0\). Here \(S_P=|a|\), and \[ E^{(j)}(a)[b_1,\ldots,b_j] =(-1)^j j!\frac{b_1\cdots b_j}{a^{j+1}}\delta_0. \] Multiplication by \(1/w_\varepsilon\) leaves \(\delta_0\) unchanged. The weight \(F_j\) is the constant \(|a|^{j+1}/\prod|b_\nu|\). Its weighted \(B_\infty\) norm times the coefficient of the delta is \(j!\), consistent with (8). For finite \(p\), a positive constant weight is not integrable on \(\mathbb R^n\) when \(n\geq1\), so the delta belongs to none of the corresponding \(B_{p,k}\) spaces with constant positive \(k\).

Exercise 2 (intermediate). On \(\mathbb R\), take \(L_a=\partial_x+a\) and its forward kernel \(E_a=1_{[0,\infty)}e^{-ax}\). Differentiate with respect to \(a\) and verify (11) at order one directly.

Solution. The derivative is \(\dot E_a=-x1_{[0,\infty)}e^{-ax}\). Since \(L_aE_a=\delta_0\), direct distributional differentiation gives \(L_a\dot E_a=-E_a\): the boundary term at zero vanishes because of the factor \(x\). Applying \(L_a\) once more gives \(L_a^2\dot E_a=-\delta_0\), which is (11) with \(Q=1\). After localization, two integrations by parts show decay of the transform of \(\dot E_a\) by \((1+|\xi|)^{-2}\); the first boundary term vanishes, while the second contains the derivative of \(-xe^{-ax}\chi(x)\) at zero. Thus the gain predicted by \(F_1=S_P^2\) is visible in this model.

Exercise 3 (intermediate). Prove that adding any distributional solution \(H\) of \(P(D)H=0\) to a fundamental solution preserves the fundamental-solution equation. Why does this observation alone give no regularity bound?

Solution. Linearity gives \(P(D)(E+H)=\delta_0\). The homogeneous equation does not force arbitrary solutions into the regular inverse scale. For example, in \(\mathbb R^2\) with \(P(\xi)=\xi_1\), take \(H=1\otimes D_2\delta_0(x_2)\). It solves \(D_1H=0\). Choose \(\chi(x)=\chi_1(x_1)\chi_2(x_2)\), with \(\chi_2=1\) near zero and nonzero \(\chi_1\). Its localized transform is \(\xi_2\widehat\chi_1(\xi_1)\). Fix a \(\xi_1\) where \(\widehat\chi_1\) is nonzero and let \(|\xi_2|\to\infty\). Multiplication by \(S_P=(1+\xi_1^2)^{1/2}\) remains unbounded. Thus adding this homogeneous solution to a regular fundamental solution destroys regularity.

Exercise 4 (advanced). Let \(P_a(\xi_1,\xi_2)=\xi_1^2+i\xi_2+a\), with \(a\) a real parameter. For a smooth family of fundamental solutions, prove that local membership of its \(j\)-th parameter derivative in \(B_{p,k}\) forces \(k/S_{P_a}^{j+1}\in L^p\). Explain why an elliptic estimate with weight \(\langle\xi\rangle^{2j+2}\) cannot be substituted.

Solution. Every parameter direction is \(Q=1\), so \(S_Q=1\) and Theorem 4.2 gives the asserted integrability. Along \(\xi_1=0\), the derivative norm is \(S_{P_a}(0,\xi_2)^2=|i\xi_2+a|^2+5\), which grows like \(\xi_2^2\), rather than \(\xi_2^4\). Therefore \(S_{P_a}^{j+1}\) has growth of order \(j+1\) in that direction. The proposed isotropic weight has growth of order \(2j+2\) and is not bounded by a constant multiple of the actual weight. The operator is not elliptic of order two, so that stronger uniform gain is unsupported.

References