Delineating the differential operators whose weak and strong $L^2$ extensions coincide

For a linear differential operator $P$ with complex-valued coefficients on an open set $\Omega \subset \mathbb{R}^n$, the weak (maximal) extension $L_w$ acts on all $L^2$ functions $u$ for which $Pu$, computed in the sense of distributions, is again square-integrable, while the strong extension $L_s$ is the closure of $P$ on compactly supported smooth functions in the graph norm. One always has $L_s \subseteq L_w$, and their identity states that test functions are dense in the maximal domain. Hörmander (1961) proved the local identity for operators of principal type with real coefficients and gave counterexamples showing that the global identity fails and that no translation-invariant mollification can establish the identity in general; the characterization for complex-valued coefficients, the complete delineation of the operator class enjoying the identity, and the minimal coefficient smoothness have remained open since. The later literature produced substantial partial results: Baiocchi (1969) built operator-adapted variants of the Friedrichs lemma but stated that translating his implicit sufficient conditions into explicit properties of the operator remains open; Bolley–Camus (1974) stated the delineation of the identity class as the general open problem and showed that non-principal-type degenerate elliptic operators can enjoy the identity; Fattorini (1979) settled the first-order case with unbounded coefficients in $\mathbb{R}^m$ sharply; Shubin (1990, from results of Sjöstrand) proved the equality for uniformly elliptic operators on manifolds of bounded geometry; and current work on Dirac operators (the first-order matrix differential operators of relativistic quantum mechanics) and on formally integrable structures (complex subbundles of the complexified tangent bundle satisfying a Frobenius-type integrability condition) still uses the 1961 first-order identity as a black box. No known work characterizes the complex-coefficient case or settles the minimal smoothness; the problem remains open.

Background

Let $\Omega$ be an open subset of $\mathbb{R}^n$ and let

$$P \;=\; \sum_{|\alpha| \le m} a_\alpha(x)\, D^\alpha$$

be a linear differential operator of order $m$ with complex-valued coefficients, where $D^\alpha$ denotes differentiation with respect to the multi-index $\alpha$, scaled by powers of the imaginary unit. Two closed realizations of $P$ as an unbounded operator on the Hilbert space $L^2(\Omega)$ of square-integrable functions are classically distinguished. The weak extension, also called the maximal operator, $L_w$, has domain consisting of all $u \in L^2(\Omega)$ such that $Pu$, computed in the sense of Schwartz distributions — that is, by moving the derivatives onto smooth compactly supported test functions through integration by parts, which requires no differentiability of $u$ — again belongs to $L^2(\Omega)$. The strong extension, $L_s$, is the closure of $P$ acting on $C_0^\infty(\Omega)$, the space of infinitely differentiable functions compactly supported in $\Omega$, taken in the graph norm $\|u\| + \|Pu\|$. One always has $L_s \subseteq L_w$, and the identity $L_s = L_w$ holds exactly when $C_0^\infty(\Omega)$ is dense in the maximal domain for the graph norm, that is, when every square-integrable $u$ with $Pu$ square-integrable can be approximated by test functions together with its image under $P$. The identity is the step by which boundary-value, regularity, and spectral questions about $P$ are reduced to questions about a concrete operator. When $P$ is formally self-adjoint — equal to the operator obtained from it by integration by parts — the identity is equivalent to essential self-adjointness, meaning that the closure of $P$ on test functions is self-adjoint and therefore generates a unitary time evolution, the property demanded of quantum-mechanical Hamiltonians.

The identity originates with Friedrichs (1944), who proved it for broad classes of operators by mollification: convolving $u$ with a smooth approximate identity — a nonnegative function of integral one in $C_0^\infty$, rescaled to a small width $\varepsilon$ — approximates $u$ in $L^2$, and the commutator estimate now called Friedrichs' lemma shows that this mollification nearly commutes with $P$, so that mollified elements of the maximal domain can be cut off and approximated by test functions. Because convolution commutes with translations, the mechanism is tied to the Euclidean geometry of $\Omega$.

The class in which Hörmander could carry the method through is the operators of principal type. The principal symbol $p_m(x,\xi) = \sum_{|\alpha| = m} a_\alpha(x) \xi^\alpha$ — the top-order part of $P$ with derivatives replaced by the components of a covector $\xi \in \mathbb{R}^n$ — encodes the geometry of the characteristics, the directions along which $p_m$ vanishes — the directions in which $P$ fails to be elliptic, ellipticity meaning that $p_m(x,\xi) \neq 0$ for every nonzero covector $\xi$. For real coefficients, principal type means that these real characteristics are simple, in that the gradient of $p_m$ with respect to $\xi$ does not vanish on the characteristic set. In Hörmander (1960) this geometric condition is equivalent to an a priori estimate controlling, for test functions supported in a compact set, the $L^2$ norms of all derivatives of $u$ of order below $m$ by $\|Pu\|_{L^2}$ together with $\|u\|_{L^2}$; the 1961 paper works with the corresponding estimate formulation, its inequality (6.1), which for real coefficients of class $C^1$ is equivalent to the simplicity of the real characteristics. Hörmander (1961) established the local identity $L_s = L_w$ for operators of principal type with real coefficients on sufficiently small open sets, together with a first-order identity for coefficients of class $C^1$ that is still routinely quoted. The same paper produced two counterexamples: a second-order wave-type operator in two variables admitting high-frequency solutions in the maximal domain that cannot be approximated by test functions in the graph norm, so that the identity fails globally; and a third-order operator in three variables for which no nonzero translation-invariant mollifier can prove the identity at all. The general theorems of that paper (Theorems 6.2 and 6.3) assume coefficients with sufficiently many continuous derivatives without estimating how many are needed.

What has survived the subsequent sixty years of work is the other half of the picture: for complex-valued coefficients no characterization of the conditions under which the estimate defining principal type — and hence the identity theorems built on it — holds was given, so the class of operators enjoying the local identity was never delineated; the delineation of this class was subsequently stated as an explicit open question by Baiocchi (1969) and as the general problem by Bolley–Camus (1974).

Problem Statement

Determine the differential operators whose weak and strong $L^2$ extensions coincide. Concretely, for a linear operator $P$ of order $m$ with complex-valued coefficients of class $C^1$ (or the weakest class in which the question is meaningful) on open sets $\Omega \subset \mathbb{R}^n$, give necessary and sufficient conditions — in terms of the principal symbol, the lower-order terms, and the coefficient regularity — under which $C_0^\infty(\Omega)$ is dense in the domain of the maximal operator for the graph norm, locally in $\Omega$, that is, on a sufficiently small neighborhood of each point. Hörmander (1961) established sufficiency for principal type with real coefficients and showed by counterexamples that the global identity fails and that operators outside principal type can fail the identity; the open parts are: (1) the complex-coefficient case, that is, necessary and sufficient conditions on the operator under which the local identity holds for complex-valued coefficients; (2) the complete delineation of the class of operators with the local identity — the "problème général" stated by Bolley–Camus (1974) and the implicit-to-explicit translation left open by Baiocchi (1969) — given as checkable conditions on the operator rather than a restatement of the identity itself, consistent, as any correct delineation must be, with the known positive classes (real principal type on sufficiently small open sets, the degenerate elliptic operators of Bolley–Camus, the first-order classes of Fattorini, uniformly elliptic operators on manifolds of bounded geometry) and with the known nonidentity examples such as Moyer (1968); and (3) the minimal number of continuous derivatives of the coefficients required by the commutator estimates behind the identity theorems, between the $C^1$ of the first-order case and the unestimated 'sufficiently many' derivatives assumed for Theorems 6.2 and 6.3 of Hörmander (1961).

References

1. Lars Hörmander, Weak and strong extensions of differential operators, Communications on Pure and Applied Mathematics 14(3) (1961) 371–379. DOI: 10.1002/cpa.3160140314, https://doi.org/10.1002/cpa.3160140314 (Zbl 0111.29202)
2. K. O. Friedrichs, The identity of weak and strong extensions of differential operators, Transactions of the American Mathematical Society 55 (1944) 132–151. DOI: 10.2307/1990143, https://doi.org/10.2307/1990143 (Zbl 0061.26201)
3. M. S. Narasimhan, The identity of the weak and strong extensions of a linear elliptic differential operator. I, II, Proceedings of the National Academy of Sciences of the United States of America 43 (1957), part I: 513–514, DOI: 10.1073/pnas.43.6.513, https://doi.org/10.1073/pnas.43.6.513; part II: 620, DOI: 10.1073/pnas.43.7.620, https://doi.org/10.1073/pnas.43.7.620 (Zbl 0081.11901)
4. Stephan Schwarz, On weak and strong extensions of partial differential operators with constant coefficients, Arkiv för Matematik 3 (1958) 515–526. DOI: 10.1007/BF02589513, https://doi.org/10.1007/BF02589513
5. Lars Hörmander, Definitions of maximal differential operators, Arkiv för Matematik 3 (1958) 501–504. DOI: 10.1007/BF02589511, https://doi.org/10.1007/BF02589511
6. Lars Hörmander, Differential operators of principal type, Mathematische Annalen 140 (1960) 124–146. DOI: 10.1007/BF01360085, https://doi.org/10.1007/BF01360085
7. K. O. Friedrichs, Symmetric positive linear differential equations, Communications on Pure and Applied Mathematics 11 (1958) 333–418. DOI: 10.1002/cpa.3160110306, https://doi.org/10.1002/cpa.3160110306
8. Leonard Sarason, On weak and strong solutions of boundary value problems, Communications on Pure and Applied Mathematics 15 (1962) 237–288. DOI: 10.1002/cpa.3160150301, https://doi.org/10.1002/cpa.3160150301
9. Robert D. Moyer, On the nonidentity of weak and strong extensions of differential operators, Proceedings of the American Mathematical Society 19(2) (1968) 487–488. DOI: 10.1090/S0002-9939-1968-0222720-6, https://doi.org/10.1090/S0002-9939-1968-0222720-6 (Zbl 0173.13204)
10. Fernando Cardoso, The identity of weak and strong extensions of pseudo-differential operators, Proceedings of the American Mathematical Society 29(1) (1971) 118–122. DOI: 10.1090/S0002-9939-1971-0275238-9, https://doi.org/10.1090/S0002-9939-1971-0275238-9 (Zbl 0231.47021)
11. Claudio Baiocchi, Définition d'opérateurs maximaux et applications, Annales scientifiques de l'École Normale Supérieure (4) 2(4) (1969) 481–520. Zbl 0197.40701, http://www.numdam.org/item?id=ASENS_1969_4_2_4_481_0
12. P. Bolley and J. Camus, Quelques propriétés des opérateurs maximaux associés à une classe d'opérateurs elliptiques et dégénérés, Annali della Scuola Normale Superiore di Pisa, Classe di Sciences (4) 1(3–4) (1974) 261–299. Zbl 0307.35045, http://www.numdam.org/item?id=ASNSP_1974_4_1_3-4_261_0
13. A. A. Dezin, Weak and strong extensions of differential operators, Mathematical Notes of the Academy of Sciences of the USSR 24 (1978) 867–873. DOI: 10.1007/BF01141546, https://doi.org/10.1007/BF01141546
14. H. O. Fattorini, Weak and strong extensions of first-order differential operators in Rm, Journal of Differential Equations 34 (1979) 353–360. DOI: 10.1016/0022-0396(79)90025-1, https://doi.org/10.1016/0022-0396(79)90025-1
15. M. A. Shubin, Appendice à l'exposé « Weak Bloch property and weight estimates for elliptic operators »: On the equality between weak and strong extensions (d'après J. Sjöstrand), Séminaire Équations aux dérivées partielles (École Polytechnique, Palaiseau), 1989–1990, exposé n° V, appendice, 10 pp. Zbl 0716.35052, http://www.numdam.org/item?id=SEDP_1989-1990____A7_0
16. Gheorghe Nenciu, Irina Nenciu, and Ryan Obermeyer, Essential self-adjointness of symmetric first-order differential systems and confinement of Dirac particles on bounded domains in $\mathbb{R}^d$, Communications in Mathematical Physics 387 (2021) 361–395. DOI: 10.1007/s00220-021-04129-4, https://doi.org/10.1007/s00220-021-04129-4 (preprint: arXiv:2010.09816, https://arxiv.org/abs/2010.09816)
17. Qingchun Ji, Jun Yao, and Guangsheng Yu, Formally integrable structures. I. Resolution of solution sheaves, Journal d'Analyse Mathématique 158 (2025) 731–759. DOI: 10.1007/s11854-025-0417-9, https://doi.org/10.1007/s11854-025-0417-9 (preprint: arXiv:2204.11176, https://arxiv.org/abs/2204.11176)