Half-space smoothness of solutions and the geometric characterization of strong P-convexity for constant-coefficient differential operators (Hörmander, 1970)

For a linear differential operator $P(D)$ with constant coefficients in $\mathbb{R}^n$ and a closed half-space $H$, Hörmander's 1970 paper on the singularities of solutions posed the condition, numbered (6.2) there, that every distribution $u$ with $P(D)u=0$ and $\operatorname{supp}u\subset H$ is a $C^\infty$ function — the absence of non-smooth null solutions supported in $H$ — and proved that (6.2) implies the algebraic condition (6.3): the principal parts of all limit polynomials of $P$, i.e. all limits of $P(\cdot+\eta)/\tilde P(\eta)$ as $|\eta|\to\infty$, are nonzero on some open cone containing the normal of $H$. Sufficiency of (6.3) was proved only for evolution operators, those admitting a fundamental solution supported in $H$; the general case was left open, together with the companion geometric question of whether strong P-convexity of an open set $\Omega$ — the property equivalent to surjectivity of $P(D)$ on the space of distributions, known to follow when every intersection $\Omega\cap(x+\Lambda'(Q))$ with the propagation subspaces of limit polynomials is convex — conversely forces that convexity. Later literature resolved the variable-coefficient parts of the program (Fourier integral operators) and the two-dimensional convexity case (Kalmes, 2011), but the two central constant-coefficient questions remain open. The problem: prove or disprove that (6.3) implies (6.2) for arbitrary constant-coefficient operators, and prove or disprove the converse of the 1970 Theorem 7.1 characterizing strong P-convexity.

Background

Let $P$ be a nonconstant polynomial in $n\ge 2$ real variables and let $P(D)$ denote the linear differential operator with constant coefficients obtained from $P$ by substituting $D_j=-i\,\partial/\partial x_j$ for the $j$-th variable. Solutions are understood in the sense of distributions, i.e. as members of the space $\mathcal{D}'(\mathbb{R}^n)$ of continuous linear functionals on the compactly supported $C^\infty$ functions. The singular support of a distribution $u$, written $\operatorname{sing\,supp}u$, is the complement of the largest open set on which $u$ coincides with a $C^\infty$ function; it records where $u$ fails to be smooth.

Hörmander's 1970 paper linked the propagation of singularities of solutions of $P(D)u=f$ to the asymptotic behavior of $P$ at infinity, as follows. For $\eta\in\mathbb{R}^n$ set $\tilde P(\eta)^2=\sum_\alpha|P^{(\alpha)}(\eta)|^2$, the sum running over all derivatives $P^{(\alpha)}$ of $P$ with respect to multiindices $\alpha$ (tuples of nonnegative integers giving the order of differentiation in each variable). A limit polynomial of $P$ is any limit, along a sequence $|\eta_\nu|\to\infty$, of the normalized translates $\theta\mapsto P(\eta_\nu+\theta)/\tilde P(\eta_\nu)$; the set $L(P)$ of all limit polynomials is a compact semi-algebraic set — a set defined by finitely many polynomial equations and inequalities — of polynomials of degree at most $\deg P$, and it encodes every degeneration of $P$ visible from infinity. For a polynomial $Q$ one defines the subspace of invariance directions $\Lambda(Q)=\{\eta:Q(\xi+t\eta)=Q(\xi)\text{ for all }t\in\mathbb{R}\text{ and all }\xi\}$ and its annihilator $\Lambda'(Q)=\{x:\langle x,\eta\rangle=0\text{ for all }\eta\in\Lambda(Q)\}$. The propagation theorems of the paper show that singularities of solutions can travel along cosets $x+\Lambda'(Q)$ with $Q\in L(P)$; when $\Lambda'(Q)$ is a line these are the analogues of the bicharacteristics, the straight lines along which singularities of hyperbolic equations propagate.

Fix a nonzero vector $N$ and the closed half-space $H=\{x:\langle x,N\rangle\ge 0\}$. The paper asks when the equation admits no non-smooth null solutions, i.e. solutions supported in $H$; such solutions are exactly what obstructs uniqueness for the Cauchy problem, the initial-value problem with data on the boundary hyperplane of $H$. The half-space smoothness condition is

$$\text{(6.2)}\qquad\text{every }u\in\mathcal{D}'(\mathbb{R}^n)\text{ with }P(D)u=0\text{ and }\operatorname{supp}u\subset H\text{ lies in }C^\infty(\mathbb{R}^n).$$

Theorem 6.1 of the paper proves that (6.2) implies the algebraic condition

$$\text{(6.3)}\qquad\text{there is an open cone }\Gamma\text{ containing }N\text{ such that }Q^0(\xi)\ne 0\text{ for all }\xi\in\Gamma\text{ and all }Q\in L(P),$$

where $Q^0$ is the principal part, i.e. the top-degree homogeneous part, of $Q$. The implication from (6.3) back to (6.2) was established in the paper only for evolution operators: $P(D)$ is called an evolution operator with respect to $H$ if it has a fundamental solution $E$ — a distribution with $P(D)E=\delta$, the Dirac distribution at the origin — with $\operatorname{supp}E\subset H$; such a fundamental solution is called regular if, for every compactly supported $C^\infty$ function $\varphi$, the Fourier transform of the product $\varphi E$ satisfies the bound $|(\varphi E)^{\wedge}(\xi)|\,\tilde P(\xi)\le C_\varphi$ for all $\xi\in\mathbb{R}^n$, with $C_\varphi$ a constant depending on $\varphi$. For evolution operators, Theorem 6.2 of the paper characterizes, by a growth condition on the complex zeros of $P$ (condition (6.7) there), when all fundamental solutions supported in $H$ are regular, and this yields $(6.3)\Rightarrow(6.2)$ in the evolution case. Whether (6.3) implies (6.2) for arbitrary constant-coefficient operators was left open, and the author announced the goal of returning to the characterization of all $P$ satisfying (6.2).

The paper's second open family concerns convexity properties of open sets that guarantee solvability. An open set $\Omega$ is called P-convex if for every compact $K\subset\Omega$ there is a compact $K'\subset\Omega$ such that every test function $\varphi\in C_0^\infty(\Omega)$ — i.e. every compactly supported $C^\infty$ function on $\Omega$ — with $\operatorname{supp}P(D)\varphi\subset K$ already has $\operatorname{supp}\varphi\subset K'$; $\Omega$ is strongly P-convex if the same holds with supports replaced by singular supports: for every compact $K\subset\Omega$ there is a compact $K'\subset\Omega$ such that every compactly supported $u\in\mathcal{E}'(\Omega)$ with $\operatorname{sing\,supp}P(D)u\subset K$ has $\operatorname{sing\,supp}u\subset K'$. By classical theorems recalled in Kalmes' 2011 account, P-convexity is equivalent to surjectivity of $P(D)$ on $C^\infty(\Omega)$ and strong P-convexity to surjectivity on $\mathcal{D}'(\Omega)$, so these geometric properties decide when the equation $P(D)u=f$ can always be solved on $\Omega$. Theorem 7.1 of the 1970 paper proves that $\Omega$ is strongly P-convex whenever every intersection $\Omega\cap(x+\Lambda'(Q))$, with $Q\in L(P)$ and $x\in\mathbb{R}^n$, is convex; Theorem 7.2 there gives four equivalent characterizations of strong P-convexity under the additional hypothesis that all the spaces $\Lambda'(Q)$ have dimension at most one; and Corollary 7.3 drew the planar consequence that every P-convex subset of $\mathbb{R}^2$ is strongly P-convex — a statement whose 1970 argument was regarded as incomplete in the later literature, and which Kalmes proved in 2011, settling a conjecture that had been open since the 1960s. The converse of Theorem 7.1, and the relation between P-convexity and strong P-convexity in dimensions three and higher, where wave-equation examples show that the two notions differ, were left open.

The variable-coefficient propagation questions raised in the same paper were settled by the Fourier integral operator literature of 1971-1972, so the surviving open core of the 1970 program consists of the two constant-coefficient characterization questions stated below.

Problem Statement

Let $P$ be a nonconstant polynomial in $n\ge 2$ real variables, and let $P(D)$ be the linear differential operator with constant coefficients obtained from $P$ by substituting $D_j=-i\,\partial/\partial x_j$ for the $j$-th variable; solutions of $P(D)u=0$ are understood in the sense of distributions, i.e. in the space $\mathcal{D}'(\mathbb{R}^n)$ of continuous linear functionals on the compactly supported $C^\infty$ functions. For $\eta\in\mathbb{R}^n$ set $\tilde P(\eta)^2=\sum_\alpha|P^{(\alpha)}(\eta)|^2$, the sum running over all derivatives $P^{(\alpha)}$ of $P$ with respect to multiindices $\alpha$ (tuples of nonnegative integers giving the order of differentiation in each variable); a limit polynomial of $P$ is any limit of the normalized translates $\theta\mapsto P(\eta_\nu+\theta)/\tilde P(\eta_\nu)$ along a sequence $|\eta_\nu|\to\infty$, and $L(P)$ denotes the set of all limit polynomials, a compact semi-algebraic set — a set defined by finitely many polynomial equations and inequalities. For a polynomial $Q$ set $\Lambda(Q)=\{\eta:Q(\xi+t\eta)=Q(\xi)\text{ for all }t\in\mathbb{R}\text{ and all }\xi\}$ and $\Lambda'(Q)=\{x:\langle x,\eta\rangle=0\text{ for all }\eta\in\Lambda(Q)\}$, and let $Q^0$ be the principal part of $Q$, i.e. its top-degree homogeneous part. Fix a nonzero vector $N$ and the closed half-space $H=\{x:\langle x,N\rangle\ge 0\}$.

The problem is the surviving open core of Hörmander's 1970 characterization program for singularities of solutions of constant-coefficient equations; it consists of two linked parts, and a complete resolution of either part at the stated generality is a full answer.

(A) Decide whether the algebraic condition

$$\text{(6.3)}\qquad\text{there is an open cone }\Gamma\text{ containing }N\text{ such that }Q^0(\xi)\ne 0\text{ for all }\xi\in\Gamma\text{ and all }Q\in L(P)$$

implies the half-space smoothness condition

$$\text{(6.2)}\qquad\text{every }u\in\mathcal{D}'(\mathbb{R}^n)\text{ with }P(D)u=0\text{ and }\operatorname{supp}u\subset H\text{ lies in }C^\infty(\mathbb{R}^n)$$

for arbitrary constant-coefficient operators. It is known that (6.2) implies (6.3) (Theorem 6.1 of the 1970 paper) and that (6.3) implies (6.2) when $P(D)$ is an evolution operator with respect to $H$, i.e. when it admits a fundamental solution — a distribution $E$ with $P(D)E=\delta$, the Dirac distribution at the origin — supported in $H$ (Theorem 6.2 there, via condition (6.7) on the complex zeros of $P$); the implication for general $P$ is the open target.

(B) In the same program, decide whether the converse of Theorem 7.1 of the 1970 paper holds. Recall that an open set $\Omega\subset\mathbb{R}^n$ is strongly P-convex for the operator $P(D)$ if for every compact $K\subset\Omega$ there is a compact $K'\subset\Omega$ such that every compactly supported $u\in\mathcal{E}'(\Omega)$ with $\operatorname{sing\,supp}P(D)u\subset K$ has $\operatorname{sing\,supp}u\subset K'$, where $\operatorname{sing\,supp}$ denotes the singular support, the complement of the largest open set on which $u$ coincides with a $C^\infty$ function; by the classical surjectivity theorems recalled in Kalmes' 2011 account, strong P-convexity is equivalent to surjectivity of $P(D)$ on $\mathcal{D}'(\Omega)$. Theorem 7.1 proves that $\Omega$ is strongly P-convex whenever every intersection $\Omega\cap(x+\Lambda'(Q))$, with $Q\in L(P)$ and $x\in\mathbb{R}^n$, is convex; the open question is whether, conversely, strong P-convexity of $\Omega$ forces convexity of all these intersections. Known partial results: Theorem 7.2 of the paper gives equivalent characterizations of strong P-convexity under the additional hypothesis that all the spaces $\Lambda'(Q)$, $Q\in L(P)$, have dimension at most one, and Kalmes (2011) proved that in the plane P-convexity — the analogous condition with the ordinary supports of test functions, i.e. for every compact $K\subset\Omega$ there is a compact $K'\subset\Omega$ such that every $\varphi\in C_0^\infty(\Omega)$ with $\operatorname{supp}P(D)\varphi\subset K$ has $\operatorname{supp}\varphi\subset K'$ — and strong P-convexity coincide for every operator; the general case is open.

Whichever part is answered is to be answered at the stated generality, with no restriction on the number of variables, the degree or structure of $P$, or the form of the answer beyond what is stated. The variable-coefficient propagation questions and the Fourier-integral-operator extension raised in the same 1970 paper were resolved by the 1971-1972 Fourier integral operator literature (Hörmander; Duistermaat and Hörmander) and are not part of the target.

References

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