Approximation and global solvability from support geometry

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

A regular fundamental solution solves an equation with compact data. Data with unrestricted growth near the boundary need a further idea. We construct solutions on increasingly large regions and correct their differences by homogeneous solutions. The geometric condition that makes this correction possible is a uniform bound on the supports of compact test functions under the transposed operator.

The prerequisites are Local regularity, sharp embeddings, and compactness, Operator strength and local inverses, the Hahn–Banach separation theorem, and the Baire theorem for Fréchet spaces. Malgrange's original paper [Malgrange] is a classical source for approximation. Hörmander [HII], Sections 10.5–10.6, supplies the theorem-specific correspondence for approximation in finite-exponent weighted spaces, the support condition, and the exhaustion argument used here. The Fourier support results used at the entry to the argument are recorded precisely below; their distribution and Fourier proofs belong to the prerequisite course.

Fix a nonzero complex polynomial \(P\), put \(D=-i\partial\), and use complex-linear distribution pairings. The formal transpose is \[ P^t=P(-D). \] For a compactly supported distribution \(v\), \(\operatorname{ch}\operatorname{supp}v\) denotes the convex hull of its support.

Continuous functionals, test families and compact limits, Section 1, proves the locally convex separation used below. The complete-metric Baire theorem and the Fréchet results are Banach estimates, quotient spaces and compact parameter arguments, Sections 6 and 14.

The compact-support Fourier inputs

An exponential-polynomial solution is a function \[ h(x)=e^{ix\cdot z}A(x),\qquad z\in\mathbb C^n, \quad A\text{ a polynomial},\qquad P(D)h=0. \tag{1} \] Equivalently, \(P(D+z)A=0\). The polynomial factor records multiplicities; the class includes more than the plane exponentials for which \(P(z)=0\).

We use the following two entry results.

Convex supports and convolution cancellation, Corollary 4.3, proves the differential support-hull equality in (2). Under our D convention the polynomial applied to ordinary derivatives is obtained by substituting minus i times each variable; this is again a nonzero complex polynomial. Empty support and the nonzero scalar case are included. The compact division and exponential-polynomial annihilator equivalence in (3) are proved in Compact Fourier division and multiplicity-sensitive annihilators, Theorems CF2.1–CF5.1, under its exact stated Fourier and Cauchy entries. CF4.1 also supplies an additional ordinary-hull proof of (2); the AN-01 Corollary 4.3 remains the other complete proof. Hörmander [HI], Theorems 7.3.1–7.3.2 and Lemma 7.3.3, gives the Paley–Wiener growth and compact-division route; Definition 7.3.5 and Lemma 7.3.7 give the exponential-polynomial annihilator argument, including polynomial multiplicities and the transpose sign. The linked L122 proof supplies the bounded division input; its lower Fourier, Cauchy, weak, elliptic and general Green provider scopes remain separate, and recursive prerequisite closure is not asserted. Malgrange [Malgrange] gives the classical approximation context.

Approximation in several regularity scales at once

Let \[ \mathcal F(X)=\bigcap_{i\in I}B_{p_i,k_i}^{\mathrm{loc}}(X), \qquad 1\leq p_i<\infty, \tag{4} \] where the \(k_i\) are moderate and \(I\ne\varnothing\). The index set may be uncountable. The topology is generated by all cutoff seminorms \(\|\chi u\|_{p_i,k_i}\).

Smooth functions belong to every space in (4), and the map from \(C^\infty(X)\) into \(\mathcal F(X)\) is continuous. Indeed each fixed weighted cutoff seminorm is bounded by finitely many uniform derivatives on a compact set. For a compactly supported member of \(\mathcal F\), convolution with a smooth approximate identity converges in every seminorm of (4). This uses \(p_i<\infty\). It applies to an uncountable family because convergence is required separately in each seminorm, and a neighborhood uses only finitely many of them.

Lemma 2.1. Let \(L\) be a continuous linear functional on \(\mathcal F(X)\). Its restriction to smooth functions is a distribution \(\mu\) with compact support in \(X\). There is a compact \(K_L\subset X\) such that \[ L((1-\chi)u)=0 \tag{5} \] whenever \(\chi\) equals one near \(K_L\). If \(P^tv=\mu\) with \(v\in\mathcal E'(X)\), then \(L(u)=0\) for every \(u\in\mathcal F(X)\) satisfying \(P(D)u=0\).

Proof. Continuity bounds \(|L(u)|\) by a finite sum of cutoff seminorms. The union of the supports of those cutoffs is a suitable \(K_L\), proving (5). The smooth restriction is continuous in the \(C^\infty\) topology and has support in \(K_L\), hence defines \(\mu\in\mathcal E'(X)\).

For the last assertion, choose \(\chi\in C_c^\infty(X)\) equal to one near \(K_L\cup\operatorname{supp}v\). Extend \(\chi u\) by zero, and mollify it to smooth functions \(u_\varepsilon\). They converge to \(\chi u\) in \(\mathcal F(X)\). For sufficiently small \(\varepsilon\), \(P(D)u_\varepsilon=0\) near \(\operatorname{supp}v\), since \(P(D)(\chi u)=0\) there and the mollifier has shrinking support. Consequently \[ L(u_\varepsilon)=\mu(u_\varepsilon) =v(P(D)u_\varepsilon)=0. \] Pass to the limit and use (5) to obtain \(L(u)=L(\chi u)=0\). This argument never pairs a distribution \(\mu\) directly with a nonsmooth \(u\). \(\square\)

Theorem 2.2. If \(X\) is convex, finite linear combinations of the solutions in (1) are dense in \(\{u\in\mathcal F(X):P(D)u=0\}\), with the topology inherited from \(\mathcal F(X)\).

Proof. Suppose a continuous \(L\) annihilates those exponential-polynomial solutions. Its smooth restriction \(\mu\in\mathcal E'(X)\) has the same annihilation property. Input (3) gives \(P^tv=\mu\) with \(v\) compactly supported in \(\mathbb R^n\). By (2), \[ \operatorname{supp}v\subset\operatorname{ch}\operatorname{supp}\mu\subset X. \] The last inclusion holds because the convex hull of a compact subset of an open convex set is still a compact subset of that set. Lemma 2.1 now says that \(L\) annihilates every homogeneous solution in \(\mathcal F(X)\). Hahn–Banach separation proves the asserted density. The solution space is closed because \(\mathcal F\) embeds continuously into distributions and distributional differentiation is continuous. \(\square\)

Here density means simultaneous approximation in any finite list of the chosen cutoff norms. If that topology is not metrizable, the assertion does not automatically provide one approximating sequence for every solution.

A support condition for Runge approximation

For an open \(Y\), write \(\mathcal E'(\overline Y)\) for global compactly supported distributions whose support is contained in the closed set \(\overline Y\). Their supports may meet its boundary. This differs from \(\mathcal E'(Y)\).

Theorem 3.1. Let \(X_1\subset X_2\) be open. Assume that \[ \begin{gathered} v\in\mathcal E'(\overline{X_2}),\quad \operatorname{supp}P^tv\subset X_1\\ \Longrightarrow\quad v\in\mathcal E'(X_1). \end{gathered} \tag{6} \] Restrictions to \(X_1\) of homogeneous solutions in \(\mathcal F(X_2)\) are dense among homogeneous solutions in \(\mathcal F(X_1)\).

Proof. Let \(L\) annihilate all such restrictions. Its smooth restriction is \(\mu\in\mathcal E'(X_1)\). Every exponential-polynomial solution lies in \(\mathcal F(X_2)\), so (3) gives a global compact \(v\) satisfying \(P^tv=\mu\). We first show \(\operatorname{supp}v\subset\overline{X_2}\).

Choose a fundamental solution \(E^t\) of \(P^t\). Compact inversion gives \(v=E^t*\mu\). Let \(\check E^t(x)=E^t(-x)\). It is a fundamental solution of \(P(D)\). For \(h\in C_c^\infty(\mathbb R^n\setminus\overline{X_2})\), the convolution pairing identity yields \[ v(h)=\mu(\check E^t*h). \tag{7} \] The function \(\check E^t*h\) is smooth and satisfies \(P(D)(\check E^t*h)=h=0\) on \(X_2\). Its restriction to \(X_2\) is therefore one of the solutions annihilated by \(L\), so the right side of (7) is zero. This proves the claimed closed-set support inclusion. Hypothesis (6) then improves it to \(v\in\mathcal E'(X_1)\). Lemma 2.1 and Hahn–Banach finish the proof. \(\square\)

The closure in (6) is essential to this proof: testing outside \(\overline{X_2}\) initially proves a closed-set inclusion, not compact support strictly inside \(X_2\).

Corollary 3.2. Suppose every global compact distribution \(v\) with \(\operatorname{supp}P^tv\subset X\) has support compactly contained in \(X\). Then exponential-polynomial solutions are dense among homogeneous solutions in \(\mathcal F(X)\).

Proof. Apply Theorem 3.1 with \(X_2=\mathbb R^n\), then apply Theorem 2.2 on \(\mathbb R^n\). Given finitely many seminorms and a tolerance, first approximate by a global solution and then approximate that solution by a finite exponential-polynomial sum, using half the tolerance at each step. \(\square\)

Convexity defined by supports

An open set \(X\) is \(P\)-convex for supports if for each compact \(K\subset X\) there is a compact \(K'\subset X\) such that \[ \phi\in C_c^\infty(X),\quad \operatorname{supp}P^t\phi\subset K \quad\Longrightarrow\quad\operatorname{supp}\phi\subset K'. \tag{8} \] This is a property of the equation and the open set together. It asks for one support bound for all test functions with the indicated image support.

Every open convex set has this property: (2) allows \(K'=\operatorname{ch}K\). We next give a distance test which also handles nonconvex sets. Write \[ d_X(A)=\inf\{|a-y|:a\in A,\ y\in\mathbb R^n\setminus X\}. \] We also write \(d_X(x)=d_X(\{x\})\) for a point. The distance to the empty complement is \(+\infty\). We exclude the zero distribution when using distances of its support; it causes no condition in (8).

Theorem 4.1. An open set \(X\) is \(P\)-convex for supports if and only if \[ d_X(\operatorname{supp}v) =d_X(\operatorname{supp}P^tv) \qquad(0\ne v\in\mathcal E'(X)). \tag{9} \]

Proof. Suppose (9) holds. For nonempty compact \(K\subset X\), put \(\delta=d_X(K)>0\). If \(\operatorname{supp}P^t\phi\subset K\), (9) puts \(\operatorname{supp}\phi\) at distance at least \(\delta\) from the complement. By (2) it also lies in \(\operatorname{ch}K\). Thus one can use \[ K'=\operatorname{ch}K\cap\{x:d_X(\{x\})\geq\delta\}, \tag{10} \] a compact subset of \(X\). For \(X=\mathbb R^n\), simply use \(\operatorname{ch}K\); for empty \(K\), injectivity gives \(\phi=0\).

Conversely, assume (8), and first take nonzero \(v\in C_c^\infty(X)\) with finite \(\delta=d_X(\operatorname{supp}v)\). Suppose the right side of (9) were greater than \(\delta\). All translates \(v_a(x)=v(x-a)\), \(|a|<\delta\), belong to \(C_c^\infty(X)\). Their transposed images have supports in the single compact subset \[ \operatorname{supp}P^tv+\overline{B(0,\delta)}\subset X. \] By (8), all supports of \(v_a\) must lie in one compact subset of \(X\). But choose a nearest pair \(x\in\operatorname{supp}v\), \(y\notin X\), \(|y-x|=\delta\). Translating by \((1-1/j)(y-x)\) moves \(x\) arbitrarily close to \(y\), a contradiction. Such a nearest pair exists by compactness of the first support and closedness of the complement. The reverse inequality in (9) follows from locality, \(\operatorname{supp}P^tv\subset\operatorname{supp}v\). If the complement is empty, both distances are infinite.

For a general nonzero \(v\in\mathcal E'(X)\), mollify it. For small \(\varepsilon\), both \(v_\varepsilon\) and \(P^tv_\varepsilon\) have compact support in \(X\), and the preceding equality applies. Their support distances converge to the distances for \(v\) and \(P^tv\). Indeed convolution enlarges each original support by at most \(\varepsilon\), giving one bound. For the other, distributional convergence prevents the mollified functions from eventually vanishing on a neighborhood of a nearest support point. Input (2) ensures \(P^tv\ne0\). Passing to the limit proves (9). \(\square\)

Corollary 4.2. Interiors of arbitrary intersections of \(P\)-convex open sets are \(P\)-convex. Directed unions are \(P\)-convex, and so is \[ \liminf{}^{\!\circ}X_j =\bigcup_N\operatorname{int}\bigcap_{j\geq N}X_j. \tag{11} \] For an arbitrary indexed family, the same conclusion holds for the open lower limit \[ \bigcup_{J\subset I\text{ finite}}\operatorname{int}\bigcap_{i\in I\setminus J}X_i. \] Every open set is contained in a smallest \(P\)-convex open set.

Proof. If \(X=\operatorname{int}\bigcap_iX_i\), its complement is the closure of the union of their complements. For \(v\in\mathcal E'(X)\), take the infimum over \(i\) of the equal distances in (9) for \(X_i\). This gives (9) for \(X\).

For a directed union \(X=\bigcup_iX_i\), each compact support lies in one \(X_i\): use a finite subcover and an upper member containing that finite subfamily. For all larger members the distance identities hold. Distances to their decreasing closed complements have supremum equal to the distance to their intersection, which is \(\mathbb R^n\setminus X\). To justify this when the supremum is finite, fix any larger radius. Each complement meets the compact closed neighborhood of the support with that radius. These intersections have the finite intersection property, so compactness supplies a point in every complement at that distance or less. Let the radius decrease to the supremum. If the supremum is infinite, the equality is immediate. Thus (9) passes to the union. Apply the intersection and directed-union conclusions to (11) and to the family indexed by finite exceptional sets \(J\).

Finally, take the interior of the intersection of all \(P\)-convex open sets containing the given open set. This family is nonempty because it contains \(\mathbb R^n\); its interior still contains the original open set. The intersection result proves the required minimality and convexity. \(\square\)

Why solvability forces this geometry

Theorem 5.1. If for every \(f\in C^\infty(X)\) there is a distributional solution of \(P(D)u=f\) in \(X\), then \(X\) is \(P\)-convex for supports.

Proof. Fix compact \(K\subset X\). Let \(G\) consist of the test functions \(\phi\in C_c^\infty(X)\) with \(\operatorname{supp}P^t\phi\subset K\), topologized by the increasing seminorms \[ q_N(\phi)=\max_{|\alpha|\leq N}\sup_{\mathbb R^n}|D^\alpha P^t\phi|. \] It is metrizable and Hausdorff by compact injectivity. Put \(B(f,\phi)=\int f\phi\), for \(f\in C^\infty(X)\). For fixed \(\phi\), this is continuous in \(f\). For fixed \(f\), choose a solution \(u\). Then \(B(f,\phi)=u(P^t\phi)\), bounded by a constant times some \(q_N(\phi)\), because the argument of \(u\) is supported in the fixed \(K\).

Here is the needed joint-continuity argument. On the Fréchet space \(C^\infty(X)\), the closed sets \[ A_l=\{f:|B(f,\phi)|\leq lq_l(\phi)\text{ for every }\phi\in G\} \] cover the space. By Baire, one \(A_l\) has interior. Taking differences of two points in that interior gives a neighborhood of zero on which \(|B(f,\phi)|\leq2lq_l(\phi)\). A smaller neighborhood is defined by a finite derivative seminorm \(p\) on a compact \(K'\subset X\). Scaling, and then passing to a limit if \(p(f)=0\), gives \[ |B(f,\phi)|\leq C p(f)q_l(\phi). \tag{12} \] Completeness of \(G\) was not needed.

If \(f\) is supported outside \(K'\), then \(p(f)=0\), so \(\int f\phi=0\) for all such \(f\). Therefore every \(\phi\in G\) has support in \(K'\). This is exactly (8). \(\square\)

The hypothesis asserts a solution for each datum. It does not assume a continuous or linear choice of solutions.

Correcting solutions on an exhaustion

We now prove the converse with precise weighted regularity. Suppose \(X\) is \(P\)-convex for supports, and let \[ \begin{aligned} \mathcal F(X)&=\bigcap_{j=1}^\infty B_{p_j,k_j}^{\mathrm{loc}}(X),\\ \mathcal G(X)&=\bigcap_{j=1}^\infty B_{p_j,k_jS_P}^{\mathrm{loc}}(X),\\ &\hspace{1em}1\leq p_j<\infty. \end{aligned} \tag{13} \] Both spaces are Fréchet, and \(P(D):\mathcal G(X)\to\mathcal F(X)\) is continuous.

For integers \(\nu\geq1\), use the exhaustion \[ X_\nu=\left\{x\in X: \begin{array}{l} |x|<\nu,\\ d_X(x)>1/\nu \end{array}\right\}. \tag{14} \] Then \(\overline{X_\nu}\subset X_{\nu+1}\), and the union is \(X\). Choose \(\phi_\nu\in C_c^\infty(X_\nu)\) equal to one near \(\overline{X_{\nu-1}}\), with \(X_0=\varnothing\). The seminorms \[ \|\phi_\mu u\|_{p_j,k_jS_P},\qquad \mu,j\geq1, \tag{15} \] generate the topology of \(\mathcal G(X)\).

The exhaustion has the support property needed for Theorem 3.1. If \(v\in\mathcal E'(\overline{X_{\nu+2}})\) and \(\operatorname{supp}P^tv\subset X_\nu\), then \(v\in\mathcal E'(X)\). Identity (9) places its support at distance strictly greater than \(1/\nu\) from the complement. Identity (2) places it in the convex hull of the image support, and hence in a ball of radius strictly less than \(\nu\). Thus \(v\in\mathcal E'(X_\nu)\).

Lemma 6.1. If \(g\in\mathcal F(X)\) vanishes on \(X_\nu\), then for any \(\varepsilon>0\) there is \(w\in\mathcal G(X)\) with \[ \begin{gathered} P(D)w=g\text{ on }X_{\nu+1},\\ \|\phi_\mu w\|_{p_j,k_jS_P}<\varepsilon \quad(\mu,j\leq\nu). \end{gathered} \tag{16} \]

Proof. The compact datum \(\phi_{\nu+2}g\), extended by zero, belongs globally to every \(B_{p_j,k_j}\). A regular fundamental solution gives one solution \(v=E*(\phi_{\nu+2}g)\) with all the local gains in (13). It is homogeneous on \(X_\nu\). By the support property just proved and Theorem 3.1, there is a homogeneous \(h\in\mathcal G(X_{\nu+2})\) such that \[ \|\phi_\mu(v-h)\|_{p_j,k_jS_P}<\varepsilon \quad(\mu,j\leq\nu). \] Take \(w=\phi_{\nu+2}(v-h)\), extending by zero. This lies in \(\mathcal G(X)\). On \(X_{\nu+1}\), the cutoff is identically one and \(P(D)v=g\), so the equation in (16) holds. On every \(\operatorname{supp}\phi_\mu\), \(\mu\leq\nu\), the same cutoff equals one, preserving the small seminorms. \(\square\)

Theorem 6.2. Every \(f\in\mathcal F(X)\) has a solution \(u\in\mathcal G(X)\) of \(P(D)u=f\).

Proof. First solve for compactly cut off data equal to \(f\) on \(X_1\), using a regular fundamental solution, and obtain \(u_1\in\mathcal G(X)\). Suppose \(u_\nu\) solves on \(X_\nu\). Its residual \(g_\nu=f-P(D)u_\nu\) belongs to \(\mathcal F(X)\) and vanishes on \(X_\nu\). Apply Lemma 6.1 with \(\varepsilon=2^{-\nu}\), and define \(u_{\nu+1}=u_\nu+w_\nu\). Then it solves on \(X_{\nu+1}\), and \[ \|\phi_\mu(u_{\nu+1}-u_\nu)\|_{p_j,k_jS_P}<2^{-\nu} \quad(\mu,j\leq\nu). \tag{17} \] For fixed \(\mu,j\), the tail of the sequence is Cauchy by summing the geometric series. Completeness gives a common \(u\in\mathcal G(X)\). Distributional continuity of \(P(D)\), applied on each exhaustion set, gives \(P(D)u=f\) throughout \(X\). \(\square\)

Every correction uses the same output weights \(k_jS_P\). The summation therefore retains the regularity gain; it does not accumulate a loss of derivatives at successive steps. The procedure is the linear-equation counterpart of the classical Mittag-Leffler construction.

Corollary 6.3. The following are equivalent:

  1. \(X\) is \(P\)-convex for supports.
  2. Every smooth datum has a smooth solution in \(X\).
  3. Every smooth datum has a distributional solution in \(X\).

Proof. For (1) implies (2), apply Theorem 6.2 with \(p_j=2\) and \(k_j=\langle\xi\rangle^j\), \(j=0,1,\ldots\). Smooth data belong to all these local spaces. The solution belongs to all local polynomial Sobolev orders because \(S_P\) has a positive lower bound. The embedding criteria already proved in the local-regularity lesson make it smooth. Assertion (2) implies (3), and Theorem 5.1 proves (3) implies (1). \(\square\)

Corollary 6.4. Every distribution of finite order on \(X\) has a solution of finite order on \(X\) when \(X\) is \(P\)-convex for supports.

Here finite order means that one derivative order works in the test-function estimate on all compact subsets; the constants may depend on the subset. The order of the solution need not equal that of the datum.

Proof. If \(f\) has order \(N\), each \(\chi f\) has Fourier growth bounded by a constant times \(\langle\xi\rangle^N\). Hence \(f\in B_{1,\langle\xi\rangle^{-M}}^{\mathrm{loc}}\) for any integer \(M>N+n\). Theorem 6.2 supplies a solution in the same local space, since its output weight is at least a positive multiple of \(\langle\xi\rangle^{-M}\). Membership there gives an order at most \(M\) on every compact subset: the distribution pairing is bounded by the weighted Fourier \(L^1\) norm times \(\sup\langle\xi\rangle^M|\widehat\psi(\xi)|\), which is bounded by finitely many derivatives of \(\psi\) of order at most \(M\) on its fixed compact support. Thus one order works throughout \(X\). \(\square\)

An arbitrary distribution can have orders that grow along an exhaustion. Corollary 6.4 alone does not establish solvability for all distributions. That question requires the additional geometry of singular supports.

Exercises with solutions

Exercise 1 (entry). For \(P(\xi)=\xi^2\) in one variable, classify the solutions in (1). Why would using only plane exponentials lose a homogeneous solution?

Solution. A solution of \(D^2u=0\) is affine, so the exponential-polynomial solutions are \(a+bx\), with \(z=0\). One can also see this from \((D+z)^2A=0\): for \(z\ne0\), the highest nonzero coefficient of a polynomial \(A\) is multiplied by \(z^2\), so no nonzero \(A\) is possible; for \(z=0\), \(A\) is affine. Plane exponentials supply only constants, since zero is the only root. They cannot approximate \(x\) in the smooth topology on an interval. The polynomial factor captures the double root.

Exercise 2 (intermediate). On a nonempty open set, let \(P=c\ne0\) be constant. Identify its support-convexity condition, homogeneous approximation statement, and global inverse.

Solution. Multiplication by \(c\) preserves the support of every distribution, so every open set is \(P\)-convex and one may take \(K'=K\). The homogeneous solution space is zero, making both density theorems trivial. The global solution is \(u=f/c\). The constant weight \(S_P=|c|\) changes each weighted norm only by a fixed scalar, consistent with Theorem 6.2.

Exercise 3 (intermediate). Suppose an exhaustion correction sequence satisfies (17). For fixed \(\mu,j\), give a quantitative tail estimate once \(r\geq\max(\mu,j)\).

Solution. For \(s>r\), the triangle inequality gives \[ \|\phi_\mu(u_s-u_r)\|_{p_j,k_jS_P} \leq\sum_{\nu=r}^{s-1}2^{-\nu}<2^{1-r}. \] Pass to the limit to obtain the same bound for \(u-u_r\). This estimates one seminorm at a time. The finite initial segment has no effect on convergence.

Exercise 4 (advanced). Explain exactly which part of the proof fails if (4) includes \(p=\infty\). Does this prove that approximation or solvability is false at that endpoint?

Solution. Lemma 2.1 extends annihilation from smooth functions by mollifying a compact distribution and taking a limit in all the chosen norms. Approximate identities need not converge in \(B_{\infty,k}\), so that step is unavailable. The same extension is needed in the Runge argument and therefore in Lemma 6.1. This identifies a missing proof mechanism; it is not a counterexample to every possible endpoint theorem. The finite-exponent assumptions in the stated results cannot be dropped on the strength of this argument.

References