Closing the sufficiency–necessity gap in Hörmander's generalized Rellich theorem for constant-coefficient operators: minimal cone hypotheses, semi-algebraicity removal, and borderline decay exponents

Hörmander (Israel J. Math. 16, 1973) extended Rellich's classical uniqueness theorem to general constant-coefficient operators $P(D)$, proving that a solution decaying in a directional, codimension-weighted $L^2$ sense inside a cone $\Gamma$ meeting the normals of the symbol's real zero-set must vanish (homogeneous case) or have compact support (inhomogeneous case), and constructing counterexamples showing the hypotheses are essentially minimal. Between the sufficient hypotheses and the necessary ones a precise gap remains: sufficiency demands an open cone containing normals at every point of the zero-set (and both normal directions at a simple real zero of each irreducible factor), while the necessity theorems only force the cone's closure to contain normals and are proved for closed semi-algebraic cones via Tarski–Seidenberg quantifier elimination; sharpness is established only in weighted-$L^2$ annulus scaling, not in the pointwise borderline sense of Rellich's $|x|^{-(n-1)/2}$ threshold. The problem is to determine the exact necessary-and-sufficient decay and cone hypotheses: (a) weaken or match the cone conditions, (b) remove the semi-algebraicity assumption from the necessity counterexamples, and (c) classify the optimal borderline decay exponents for all operator classes. The three parts are independent: part (c) may be answered for the cones admissible under the currently-known sufficient hypotheses without waiting on part (a).

Background

For a linear partial differential operator with constant coefficients, written $P(D)$ with $D_j=-i\,\partial/\partial x_j$ and polynomial symbol $P(\xi)$, the Fourier transform converts the equation $P(D)u=0$ into the statement that the distribution $\hat u$ is supported in the real characteristic set $A=\{\xi\in\mathbb{R}^n:P(\xi)=0\}$, the real zero-set of the symbol. This set is a real analytic set, hence a locally finite union of analytic manifolds of possibly different codimensions (codimension = dimension of the ambient space minus dimension of the manifold; the codimension of $A$ is the smallest of these). A classical theorem of Rellich (1943) is the model case: if $u$ solves the Helmholtz equation $(\Delta+k^2)u=0$ outside a ball and $u(x)=o(|x|^{-(n-1)/2})$ as $|x|\to\infty$, then $u\equiv 0$; the exponent $(n-1)/2$ is sharp, and this uniqueness-at-infinity underlies the absence of embedded eigenvalues (eigenvalues of the operator lying inside its continuous spectrum, corresponding to bound states buried among the scattering states) and the radiation conditions of scattering theory (outgoing-wave asymptotics, in the sense of Sommerfeld, that single out the physically relevant solution). Trèves (1960) studied decay of differential polynomials, and Littman (1970) proved decay-at-infinity theorems under curvature assumptions on the characteristic set. Hörmander (1973) removed such restrictions for general constant-coefficient operators. His results are naturally stated with a cone $\Gamma\subset\mathbb{R}^n$ (a set invariant under positive scalings), required open for the sufficiency theorems and permitted closed for the necessity counterexamples, and the annuli $\Gamma_R=\{x\in\Gamma:R<|x|<2R\}$. A normal vector to a manifold $M\subset A$ at a point $\xi_0$ is a nonzero vector orthogonal to the tangent space of $M$ at $\xi_0$, and a normal plane is the linear span of such normals; a cone intersects a normal plane if it contains one of its nonzero vectors. Writing $k$ for the codimension of $A$, Hörmander's homogeneous theorem states: if $\Gamma$ is an open cone intersecting each normal plane of every analytic manifold contained in $A$, and $u\in\mathscr{S}'\cap L^2_{\mathrm{loc}}$ (tempered, locally square-integrable) satisfies $P(D)u=0$ and $\lim_{R\to\infty}R^{-k}\int_{\Gamma_R}|u(x)|^2\,dx=0$, then $u=0$. The inhomogeneous version factors $P=c\prod_j P_j^{m_j}$ into real irreducible polynomial factors and assumes each $P_j$ has a simple real zero, i.e. a real point $\xi^j$ with $P_j(\xi^j)=0$ and $\nabla P_j(\xi^j)\neq 0$; if $\Gamma$ contains both $\nabla P_j(\xi^j)$ and its negative at such zeros (in addition to the normal-plane intersection property for all of $A$), then a solution of $P(D)u=f\in\mathscr{E}'$ ($f$ compactly supported) with $\varliminf_{R\to\infty}R^{-1}\int_{\Gamma_R}|u(x)|^2\,dx=0$ — here $\varliminf$ is the limit inferior, the smallest subsequential limit of the preceding quantity — must itself have compact support; an auxiliary $L^q$-criterion gives compact support whenever $u\in L^q(\mathbb{R}^n)$ with $q\le 2n/(n-1)$. In the opposite direction Hörmander proved necessity: if a decay exponent $N$ and a cone do force vanishing, then the closure of $\Gamma$ must contain a normal at every point of every manifold in $A$ and $N\le\operatorname{codim}M$; and if some irreducible factor of $P$ has no simple real zero, then for every $N$ there exist solutions of $P(D)u=f\in\mathscr{E}'$ with $u(x)=o(|x|^{-N})$ that are not compactly supported. For the directional cone hypothesis of the compactness theorem the necessity counterexamples — which show that a cone containing only one of the two normal directions at each real zero cannot support the conclusion — were constructed only for closed semi-algebraic cones $\Gamma$ (sets defined by finitely many real polynomial equations and inequalities), the estimates relying on the Tarski–Seidenberg theorem, i.e. quantifier elimination for such sets, which supplies the required uniform algebraic bounds. The published abstract claims the hypotheses on $\Gamma$ and on $P$ are minimal, and this is true in the coarse sense just described; the fine borderline between the sufficient and the necessary conditions was, however, left open, and the pointwise analogue of Rellich's sharp threshold $(n-1)/2$ was not classified for general operators. Later work extended the setting rather than the sharpness: Agmon–Hörmander (1976) developed the corresponding asymptotic theory for operators with simple characteristics (operators whose symbol has nonzero gradient at every point of its real zero-set, so that the characteristic set is a smooth surface); Murata–Shibata (1978) treated the exterior of a closed proper convex cone, keeping the same cone-hypothesis structure verbatim; Shibata (1981, 1982) treated unbounded domains and half-spaces; Littman–Yan (1996) treated right-hand sides supported in infinite cylinders; Vesalainen (2014) proved Rellich-type theorems in different decay regimes (super-exponential decay, exponentially thin supports, discrete Laplacians); Hitrik–Krupchyk–Ola–Päivärinta (2010) and the subsequent transmission-eigenvalue and non-scattering literature (transmission eigenvalues are the exceptional energies at which an inhomogeneous scattering obstacle admits an incident field producing no contrast in the far field; non-scattering energies are those at which the obstacle is invisible in the far field) — e.g. Kow–Salo–Shahgholian 2023 — use Hörmander's Corollary 3.2 as a black-box 'generalized Rellich theorem'.

Problem Statement

Determine the sharp form of Hörmander's 1973 generalized Rellich theorem for constant-coefficient operators: close the gap between the sufficient and the necessary hypotheses, so that the decay required to force a solution of $P(D)u=0$ (or of $P(D)u=f\in\mathscr{E}'$ with the compact-support conclusion) to vanish or to have compact support is characterized by necessary and sufficient conditions. Concretely, for a constant-coefficient operator $P(D)$ on $\mathbb{R}^n$ with real characteristic set $A=\{\xi\in\mathbb{R}^n:P(\xi)=0\}$ of codimension $k>0$: (a) Cone hypotheses, sufficiency side. The known sufficient conditions require a single open cone $\Gamma$ intersecting each normal plane of every analytic manifold contained in $A$, and — for the compact-support conclusion — containing both $\nabla P_j(\xi^j)$ and $-\nabla P_j(\xi^j)$ at a simple real zero $\xi^j$ of each real irreducible factor $P_j$ of $P$; the known necessity results only force the closure $\overline{\Gamma}$ to contain a normal at every point of every such manifold and the exponent to satisfy $N\le\operatorname{codim}M$. Decide exactly which cones are admissible: can the openness, the globality (one uniform cone for all points of the zero-set), or the two-sided normal condition be weakened — e.g. to cones whose closure merely meets the normal planes, to boundary or limiting cones, or to locally admissible cone families — without destroying the implication $\lim_{R\to\infty}R^{-k}\int_{\Gamma_R}|u(x)|^2\,dx=0$, $P(D)u=0$ $\Rightarrow$ $u=0$, or the corresponding compact-support implication with the exponent tied to codimension one? (b) Cone hypotheses, necessity side. Hörmander's counterexamples showing that a cone containing only one of the two normal directions at each real zero cannot support the compact-support conclusion are proved for closed semi-algebraic cones $\Gamma$, the estimates relying on Tarski–Seidenberg quantifier elimination. Prove the same necessity for arbitrary closed cones, or identify the weakest explicit regularity hypothesis on $\Gamma$ under which one-sided cones admit rapidly decaying, non-compactly-supported solutions; dually, exhibit any non-semi-algebraic one-sided cone for which the compact-support conclusion nevertheless fails or holds. (c) Borderline decay exponents. The sharpness of the 1973 theorem is in the codimension-weighted $L^2$ annulus scaling $R^{-k}$; classify the optimal borderline decay — in the pointwise sense of Rellich's $|x|^{-(n-1)/2}$ threshold for the Helmholtz operator, or in a sharpened integral sense — for all constant-coefficient operator classes: for which operators $P$ and exponents does decay $u(x)=O(|x|^{-N})$ (or $o$) inside an admissible cone — that is, a cone satisfying the sufficient hypotheses of Hörmander's theorem as currently known, or one of the weakened cone classes considered in part (a) — force $u=0$ resp. compact support, and which borderline exponents admit nontrivial decaying solutions?

References

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