An explicit algebraic or geometric characterization of Hörmander's subelliptic test estimate for first-order systems beyond the two-dimensional case

Hörmander's 1980 paper "On the subelliptic test estimates" determined when the a priori estimate $\int|u|^2\,dx \le C\int|L(x,D)u|^2\,dx$ holds for first-order systems $L(x,D)=A(x)+B(D)$ acting on compactly supported smooth functions with values in a finite-dimensional complex vector space. This "test estimate" is the local model behind subellipticity criteria for systems of pseudodifferential operators. Theorem 7.4 gives a necessary and sufficient condition: effective injectivity, on the Schwartz space, of a family of reduced operators indexed by the closure $\mathcal G_L$ of the graphs of the symbol in a Grassmannian. The condition is not explicit — it requires checking a large family of limit operators whose injectivity is itself hard to verify — and only the case $\dim W_1=2$ (Theorem 8.1) was made explicit. The open problem is to find an algebraic or geometric characterization, expressible directly in terms of the data $(A,B)$ and verifiable in practice, for general $\dim W_1$.

Background

A linear partial differential operator $P$ is called hypoelliptic if every distribution solution of $Pu=f$ is smooth wherever $f$ is smooth, and subelliptic if, roughly, it gains some positive fraction of a derivative in such regularity conclusions — certified by a priori estimates comparing Sobolev norms (norms measuring the size of derivatives of $u$) of $u$ and $Pu$. For scalar operators the theory is largely algebraic, but for systems — operators acting on vector-valued functions, whose symbols are matrices — invertibility of the symbol fails along sets that a single scalar equation cannot describe, and no comparably explicit criterion is known.

The model problem is studied in Hörmander's paper "On the subelliptic test estimates" (1980). Let $V$ be a finite-dimensional real vector space with dual $V'$, let $W_1,W_2$ be finite-dimensional complex vector spaces, and let $A:V\to L(W_1,W_2)$ and $B:V'\to L(W_1,W_2)$ be linear maps into the space of linear maps from $W_1$ to $W_2$. With $D=-i\partial/\partial x$, the first-order system $L(x,D)=A(x)+B(D)$ acts from $C_0^\infty(V,W_1)$ (smooth, compactly supported, $W_1$-valued functions) to $C^\infty(V,W_2)$. The test estimate is

$$\int_V |u|^2\,dx \le C\int_V |L(x,D)u|^2\,dx \quad \text{for all } u\in C_0^\infty(V,W_1).$$

It is called a test estimate because it is the elementary case that every criterion for subellipticity of systems of pseudodifferential operators (operators of the form $Pu(x)=(2\pi)^{-n}\int e^{i\langle x,\xi\rangle}p(x,\xi)\hat u(\xi)\,d\xi$ with an amplitude $p$ satisfying standard derivative estimates) must ultimately reproduce: after blowing up the symbol of a general system at a characteristic point, its first non-trivial Taylor terms have exactly the linear form $L(x,\xi)=Ax+B\xi$.

The paper's main result, Theorem 7.4, characterizes the estimate as follows. Let $\mathcal G_L$ be the closure, in the Grassmannian (the manifold of linear subspaces of a given dimension) of $\dim W_1$-planes in $W_1\oplus W_2$, of the set of graphs $\{(w,L(y,\eta)w): w\in W_1\}$ as $(y,\eta)$ ranges over phase space $V\oplus V'$. For each limit plane $G\in\mathcal G_L$ one reads off a projected operator $P_G L(x,D)+B_G$ (the operator induced on the limit plane $G$ by the symbol data of $L$; the precise construction is given in the source paper), and the estimate holds if and only if each of these is effectively injective on the Schwartz space $\mathcal S$ (the smooth functions all of whose derivatives decay faster than any power). Effective injectivity means: if $N=\{(y,\eta): L(y,\eta)=0\}$ is the null space of the linear symbol and $\sigma(x,\xi;y,\eta)=\langle y,\xi\rangle-\langle x,\eta\rangle$ is the canonical symplectic (anti-symmetric, non-degenerate) form on $V\oplus V'$, then on the reduced space $\tilde V=N^\sigma/(N\cap N^\sigma)$ ($N^\sigma$ being the symplectic orthogonal of $N$) the operator induced by $L$, with any constant term adjoined, is injective on $\mathcal S(\tilde V)$. This is necessary and sufficient but not explicit: $\mathcal G_L$ is an infinite set of limit planes, and each injectivity test is itself a nontrivial spectral problem.

For $\dim W_1=2$, Theorem 8.1 reduces the criterion to two conditions that can be checked directly: (a) a rank-one image condition — whenever $L(y,\eta)$ has rank 1, the image under $L(x,\xi)$ of its kernel must fail to lie in its range for suitable $(x,\xi)$ in a positive Lagrangian plane (a complex subspace $\lambda$ that is maximal isotropic for $\sigma$ and satisfies $\sigma(s,\bar s)/i\ge 0$ for $s\in\lambda$); and (b) effective injectivity of $L(x,D)$ itself. Example 8.2 then exhibits, for $V=\mathbb R^2$ and $W_1=W_2=\mathbb C^2$, a two-by-two triangular system for which the estimate holds but is destroyed by an arbitrarily small off-diagonal perturbation: the set of admissible $L$ is not open when $\dim W_1>1$ and $\dim V>1$. Theorem 9.1 converts the test-estimate criterion into a characterization of subellipticity for pseudodifferential systems with principal symbol homogeneous of order $1/2$, which is the principal application.

Two adjacent research programs frame the open question. First, the maximal-hypoellipticity program for systems, announced in Nourrigat's 1986–87 séminaire exposé "Systèmes sous-elliptiques" and continued in Nourrigat (1986, 1991): a priori estimates for systems of pseudodifferential operators are characterized by injectivity, on the Schwartz space, of limit systems, which can be interpreted through irreducible unitary representations of nilpotent Lie groups — an algebraic, representation-theoretic criterion, but established for systems whose symbols are polynomial in a family of vector fields, a different class from the general linear test data $(A,B)$. Second, the study of Melin-type lower bounds for systems, surveyed by Brummelhuis (2001): the scalar Melin inequality (Melin 1971) is the sharp local lower bound for operators with non-negative principal symbol, and its extension to systems leads to matrix oscillators (second-order operators with matrix-valued quadratic symbols, also called harmonic oscillator systems), whose spectra are explicitly computable only in special cases. Further neighbors include subelliptic estimates for systems of complex vector fields of the form $\partial_{t_j}+i(\partial\varphi/\partial t_j)\partial_x$ (Derridj–Helffer 2008), where hypoellipticity does not force subellipticity, and the pseudospectral analysis of operators with double characteristics (Parmeggiani–Pravda-Starov 2014). None of these lines produces an explicit criterion for the original test estimate.

Problem Statement

Let $V$ be a finite-dimensional real vector space with dual $V'$, let $W_1,W_2$ be finite-dimensional complex vector spaces, and let $A:V\to L(W_1,W_2)$ and $B:V'\to L(W_1,W_2)$ be linear maps. Set $L(x,\xi)=Ax+B\xi$ and let $L(x,D)=A(x)+B(D)$ with $D=-i\partial/\partial x$ act on $C_0^\infty(V,W_1)$. The test estimate is

$$\int_V |u|^2\,dx \le C\int_V |L(x,D)u|^2\,dx \quad \text{for all } u\in C_0^\infty(V,W_1).$$

Hörmander's Theorem 7.4 characterizes this estimate by effective injectivity, on the Schwartz space, of the operators $P_G L(x,D)+B_G$ for every plane $G$ in the limit-graph set $\mathcal G_L$, and Theorem 8.1 makes the criterion explicit when $\dim W_1=2$. The problem is to find, for general $\dim W_1$ (and arbitrary $\dim V$), an algebraic or geometric characterization of the validity of the test estimate that (i) is expressed directly in terms of the maps $A$ and $B$ (or equivalent finite-dimensional data, such as the structure of the limit-graph set $\mathcal G_L$ itself), and (ii) can be verified in practice — that is, without quantifying over the full set $\mathcal G_L$ and without testing injectivity on the Schwartz space of infinitely many reduced operators.

References

1. Lars Hörmander, On the subelliptic test estimates, Communications on Pure and Applied Mathematics 33(3), 339–363 (1980). DOI: 10.1002/cpa.3160330307, https://doi.org/10.1002/cpa.3160330307 (zbMATH Zbl 0414.35078)
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