Open ends of Hörmander's 1955 operator-theoretic program for general partial differential operators: in-the-large solvability, curved-boundary Cauchy spaces, and the fate of the comparison, spectral-asymptotic, and weak-strong extension questions

Lars Hörmander's 1955 Acta Mathematica paper 'On the theory of general partial differential operators' introduced the L² minimal/maximal operator framework, the strength comparison of differential operators, the Cauchy space, operators of local type, and principal type, and explicitly left five issues open: (1) extension of the weaker-operator characterization (Theorem 2.2) from bounded to unbounded domains; (2) boundary regularity and Cauchy-space structure at curved (non-planar) boundaries, treated in 1955 only at plane boundary portions (Theorem 2.13); (3) sharp spectral-function asymptotics for self-adjoint operators of local type, blocked then by the insufficiency of Ganelius' Tauberian theorem; (4) solvability for variable-coefficient principal-type operators, established in Theorem 4.1 only locally ('not valid in the large'); and (5) whether the maximal operator P is the closure of its restriction to smooth functions. The five issues have the following later careers: issue (3) was substantially resolved by Nilsson (1965, 1972) using precisely Ganelius' Stieltjes-transform Tauberian theorem; issue (5) was settled positively for smooth-coefficient classes (Friedrichs, Hörmander 1961, Cardoso, Fattorini) and negatively for singular-coefficient and boundary-condition realizations (Phillips–Sarason, Moyer 1968, which records Ralston's unpublished example, and Sarason 1984); issue (4)'s local part was superseded by the modern solvability theory culminating in the resolution of the Nirenberg–Treves conjecture (Dencker 2006), while its global part survives as the active field of global solvability/hypoellipticity (Bergamasco school, through 2026); issue (1) was subsumed by the P-convexity and comparison framework of Hörmander's ALPDO II; issue (2) found no direct follow-up and remains the least settled. The surviving open core is the in-the-large (global) solvability problem and the curved-boundary Cauchy-space structure; the resolved components are documented as verifiable sub-answers rather than discarded.

Background

Let $\Omega\subset\mathbb{R}^\nu$ be a domain and $p$ a linear differential operator. Hörmander (1955) studies $p$ through two unbounded operators in $L^2(\Omega)$: the minimal operator $P_0$, the closure of $p$ acting on $C_0^\infty(\Omega)$, and the maximal operator $P$ (the adjoint of the minimal operator of the formal adjoint), whose domain $\mathcal{D}_P$ consists of all $u\in L^2(\Omega)$ with $pu\in L^2(\Omega)$ in the sense of distributions. $P$ is closed, so $P_0\subset P$. The operator $Q$ is called weaker than $p$ (Definition 1.2) if $\mathcal{D}_{P_0}\subset\mathcal{D}_{Q_0}$, i.e. if every function in the minimal domain of $p$ is automatically in the minimal domain of $q$; for a bounded domain and constant coefficients, Theorem 2.2 characterizes this by the algebraic inequality $|Q(\xi)|\le C\,\tilde P(\xi)$ for all real $\xi$, where $\tilde P(\xi)^2=\sum_\alpha|P^{(\alpha)}(\xi)|^2$ and $P^{(\alpha)}=\partial^\alpha P$. The quotient $C=\mathcal{D}_P/\mathcal{D}_{P_0}$ with the quotient norm is the Cauchy space (Definition 1.3): its elements are the boundary data carried by functions of $\mathcal{D}_P$, and boundary problems are posed as the choice of a subspace $B\subset C$. An operator with constant coefficients is of local type (Definition 3.1) if $\mathcal{D}_P$ is stable under multiplication by cutoffs in $C_0^\infty$; for such operators Hörmander constructed a fundamental solution and proved Weyl-type regularity (in the sense of Weyl's classical lemma for the Laplacian: solutions of $Pu=0$ are automatically smooth). An operator with variable coefficients is of principal type if the partial derivatives $\partial p/\partial\xi_k$ of its principal part do not vanish simultaneously for real $\xi\ne 0$. The spectral function of a self-adjoint realization is the Carleman kernel $e_\lambda(x,y)$ of its spectral resolution $E(\lambda)$, and its asymptotics as $\lambda\to+\infty$ classically require a Tauberian theorem (a limit theorem converting weighted means of a measure into pointwise asymptotics of its distribution function). Within this framework the paper proved, among other things, the local solvability of real principal-type operators (Theorem 4.1, in a sufficiently small neighbourhood), and closed with the five gaps listed in the abstract, each phrased by the author as an admitted incompleteness rather than a formally posed conjecture. The importance of the bundle is historical as well as mathematical: it is the seed of the modern theories of strength and hypoellipticity (an operator is hypoelliptic if $u$ is smooth wherever $pu$ is smooth), of non-homogeneous boundary value problems, of spectral asymptotics for non-elliptic operators (an operator is elliptic when its principal symbol vanishes only at $\xi=0$), and of the solvability program that led from Lewy's counterexample (1957, a linear partial differential equation with smooth coefficients that has no solution even locally) to the Nirenberg–Treves conjecture — that local solvability of a principal-type operator is equivalent to condition $(\psi)$, i.e. to the principal symbol's imaginary part not changing sign from $+$ to $-$ along the bicharacteristics (the flow curves of the principal symbol's real part) — and its resolution (2006).

Problem Statement

This problem asks for the completion of the unresolved components of Hörmander's 1955 program. (4) In-the-large solvability: Theorem 4.1 proves, for a real principal-type operator with continuously differentiable principal coefficients, an a priori estimate $\sum_{|\alpha|\le m-1}\|D^\alpha u\|^2\le C\|Pu\|^2$ and local solvability only in a sufficiently small neighbourhood of a point, the author stating the results 'are not valid in the large' (in the small, i.e. near one point, versus in the large, i.e. globally on the whole domain). Characterize, by geometric and/or analytic conditions on the operator and the domain, exactly when the a priori estimate and the solvability of $Pu=f$ for every $f\in L^2(\Omega)$ — that is, the surjectivity of the maximal realization $P:\mathcal{D}_P\to L^2(\Omega)$ — hold globally (the necessary-and-sufficient criterion sought by the modern descendant of the problem, for a (pseudo)differential operator — the generalization of differential operators obtained by letting a general smooth symbol act as a Fourier multiplier — on a compact or non-compact manifold, a program pursued for vector fields on tori and related structures by the Bergamasco school since the 1990s, with no general criterion known). (2) Curved-boundary Cauchy spaces: Theorem 2.13 characterizes, at a point of a plane boundary portion, when an $L^2$ function belongs locally to the minimal domain, via square integrability of the derivatives $P^{(\alpha)}(D)u$ and strong vanishing of their traces on parallel surfaces. Extend the local characterization of the minimal domain, and correspondingly the structure of the Cauchy space $C=\mathcal{D}_P/\mathcal{D}_{P_0}$ as a space of boundary data, to points of a general curved boundary for general operators, clarifying the relation to the trace theorems of the Lions–Magenes theory. (1) the extension of Theorem 2.2 to unbounded domains (subsumed by the comparison and P-convexity framework of Hörmander's treatise The Analysis of Linear Partial Differential Operators II (1983), where P-convexity denotes the geometric condition on a domain — characterized by the impossibility of trapping compactly supported solutions with singular support — that, for operators with constant coefficients, is necessary and sufficient for solving $Pu=f$ for every $f$, whereas for the variable-coefficient principal-type operators of issue (4) no such general criterion is known); (3) the sharp spectral-function asymptotics for self-adjoint operators of local type (substantially resolved by Nilsson 1965, 1972 via Ganelius' Tauberian theorem); and (5) the identity of the maximal operator with the closure of its restriction to smooth functions (positive for smooth-coefficient classes since Friedrichs 1944 and Hörmander 1961; nonidentity in the singular-coefficient and boundary-condition settings per Phillips–Sarason 1966 and Moyer 1968, which records Ralston's unpublished example).

References

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