Hörmander (1958) extended the regularity theory of constant-coefficient hypoelliptic operators to systems: he characterized hypoelliptic systems by a distance condition on the rank-deficiency variety of the symbol matrix, proved directional Gevrey estimates ∣⟨y,D⟩μu(x)∣≤Cμ+1Γ(ρ(y)μ)\lvert\langle y,D\rangle^\mu u(x)\rvert \le C^{\mu+1}\Gamma(\rho(y)\mu) for solutions together with their optimality, and showed that the exponent ρ(y)\rho(y) takes finitely many rational values, constant on the strata of a flag of subspaces. The paper explicitly declines to work out the analogous estimates for mixed directional derivatives — the counterpart of Gevrey's classical heat-equation bounds — calling them 'very easy' but leaving them unproved, and it notes without proof that the estimates extend beyond hypoelliptic systems to any case where the exponent ρ(y)\rho(y) exists. The scalar counterpart of these estimates has been established (Hörmander's 1983 monograph and the multianisotropic Gevrey literature, e.g. Bouzar–Dali 2009 and Calvo–Hakobyan 2005), and partial hypoellipticity for systems has been developed at the C∞C^\infty level (Gårding–Malgrange 1961; Matsuura 1962); however, no published treatment is known of complete mixed-derivative Gevrey estimates for hypoelliptic constant-coefficient systems, of their optimality, or of the ρ(y)\rho(y)-extension to non-hypoelliptic systems. The problem of stating and proving these estimates in full therefore remains open.

Background

For a linear partial differential operator P(D)P(D) with constant coefficients — a polynomial in the derivatives Dj=−i ∂/∂xjD_j = -i\,\partial/\partial x_j acting on functions or distributions of x=(x1,…,xν)x=(x_1,\dots,x_\nu) — the fundamental regularity question is whether all distributional solutions of P(D)u=fP(D)u=f are automatically infinitely differentiable wherever ff is; such an operator is called hypoelliptic. (A distribution is a continuous linear functional on the space of smooth compactly supported test functions; it is the standard framework in which weak solutions live.) Hörmander (1955) characterized hypoelliptic polynomials completely: P(D)P(D) is hypoelliptic if and only if the distance from the real points ξ∈Rν\xi\in\mathbb{R}^\nu to the complex zero set V={ζ∈Cν:P(ζ)=0}V=\{\zeta\in\mathbb{C}^\nu : P(\zeta)=0\} tends to infinity as ∣ξ∣→∞\lvert\xi\rvert\to\infty, and for hypoelliptic equations he also proved quantitative bounds on the growth of derivatives of solutions. The natural scale for such bounds is the Gevrey scale: a function is of Gevrey class s≥1s\ge 1 if its derivatives satisfy ∣∂αu∣≤C∣α∣+1(α!)s\lvert\partial^\alpha u\rvert \le C^{|\alpha|+1}(\alpha!)^s on compact sets; class 11 is real analyticity, and Gevrey (1918) had shown that solutions of the heat equation are of class 22 in the time variable and analytic (class 11) in the space variables, with classical bounds also for the mixed time-space derivatives — the prototype of direction-dependent (anisotropic) regularity. When several Gevrey orders coexist, one for each of several groups of directions, the regularity is called multi-anisotropic (also multianisotropic); estimates for a derivative involving several directions at once, for example a mixed time-space derivative, are called mixed-derivative estimates.

The present paper (Hörmander, Arkiv för Matematik 3, 1958) carries this theory to systems P(D)u=f\mathcal{P}(D)u=f, where u=(u1,…,un)u=(u_1,\dots,u_n), f=(f1,…,fm)f=(f_1,\dots,f_m), and P\mathcal{P} is an m×nm\times n matrix of constant-coefficient operators; when m>nm>n the system is overdetermined (more equations than unknowns), which is the case requiring new ideas. The system is hypoelliptic if the vector distribution uu is C∞C^\infty in every open set where ff is. The paper's results are: (i) the characterization — the system is hypoelliptic if and only if the distance from real points to the rank-deficiency variety V={ζ∈Cν:rank⁡P(ζ)<n}V=\{\zeta\in\mathbb{C}^\nu : \operatorname{rank}\mathcal{P}(\zeta)<n\} tends to infinity (Theorems 1–2 and their corollary); (ii) derivative estimates (Theorem 3) — if uu solves the homogeneous system P(D)u=0\mathcal{P}(D)u=0, then for every real direction yy and every compact KK in the domain,

∣⟨y,D⟩μu(x)∣≤Cμ+1Γ(ρ(y) μ),x∈K, μ=0,1,2,…\lvert\langle y,D\rangle^\mu u(x)\rvert \le C^{\mu+1}\Gamma(\rho(y)\,\mu),\qquad x\in K,\ \mu=0,1,2,\dots

where Γ\Gamma is the Gamma function (so the bound is Gevrey of order ρ(y)\rho(y) in the direction yy) and the exponent ρ(y)≥1\rho(y)\ge 1 is read off from the geometry of VV: if M(τ)M(\tau) denotes the distance from the real points ξ\xi with ∣⟨y,ξ⟩∣=τ\lvert\langle y,\xi\rangle\rvert=\tau to VV, then M(τ)=aτb(1+o(1))M(\tau)=a\tau^b(1+o(1)) with 0<b≤10<b\le 1 and ρ(y)=1/b\rho(y)=1/b; (iii) optimality (Theorem 4) — by a Baire category argument (a completeness argument producing a worst-case solution), any sequence bound valid for all solutions must grow at least like Γ(ρ(y)μ)\Gamma(\rho(y)\mu), up to the size of the constants; and (iv) structure (Theorem 5) — there is a strictly increasing flag of subspaces {0}=G0⊂G1⊂⋯⊂Gk=Rν\{0\}=G_0\subset G_1\subset\dots\subset G_k=\mathbb{R}^\nu and a strictly increasing sequence of rational numbers 1≤σ1<⋯<σk1\le\sigma_1<\dots<\sigma_k such that ρ(y)=σj\rho(y)=\sigma_j whenever jj is the smallest index with y∈Gjy\in G_j, so that ρ\rho is constant on each stratum Gj∖Gj−1G_j\setminus G_{j-1}; in particular ρ≡1\rho\equiv 1 exactly for elliptic systems (those with only analytic solutions), and for the heat equation ρ=2\rho=2 in the time direction and 11 in the space directions, so Theorems 3 and 5 generalize Gevrey's classical heat-equation results, as the paper explicitly notes.

Two technical points recur in what follows. First, the proofs reduce the system to scalar equations: if QQ is a determinant formed from nn rows of the matrix P\mathcal{P}, an adjugate identity shows that every component uju_j of a solution of the homogeneous system satisfies Q(D)uj=0Q(D)u_j=0, and more generally R(D)uj=0R(D)u_j=0 for every RR in the ideal generated by these determinants; a theorem of Lech (1958) — a metric property of the zeros of a complex polynomial ideal, asserting that some single element RR of the ideal has a zero set metrically comparable with the common zero set of the whole ideal — then allows the scalar theory to be applied. Second, when a system is not hypoelliptic, solutions of the homogeneous system need not be C∞C^\infty, so any extension of the estimates to such systems must first specify for which class of solutions, or in which variables, the statement is made; the relevant later theory is partial hypoellipticity, where solutions are smooth in one group of variables once smoothness there is imposed.

Problem Statement

The source paper proves directional Gevrey estimates for a single direction, states their optimality, and determines the flag structure of the exponent ρ(y)\rho(y), but it does not work out the estimates for mixed directional derivatives, noting only that this 'is very easy' and referring to the estimates 'such as are known for the equation of heat'. The problem is to work out these estimates completely, and to extend the estimates beyond the hypoelliptic class, as follows. Both parts below are required for a complete resolution; within part (1), the one-direction estimate of Theorem 3 and the heat equation are special cases that the general mixed estimate must reproduce, rather than independent objectives.

(1) Mixed directional derivative estimates for hypoelliptic systems. Let P(D)\mathcal{P}(D) be a hypoelliptic system with constant coefficients (an m×nm\times n matrix of operators, m≥nm\ge n) and let uu be a C∞C^\infty solution of the homogeneous system P(D)u=0\mathcal{P}(D)u=0 in an open set Ω⊂Rν\Omega\subset\mathbb{R}^\nu. State and prove complete estimates for arbitrary mixed directional derivatives ⟨y1,D⟩μ1⋯⟨yr,D⟩μru\langle y_1,D\rangle^{\mu_1}\cdots\langle y_r,D\rangle^{\mu_r}u on compact subsets of Ω\Omega, with the exponent and weight structure determined by the flag {0}=G0⊂⋯⊂Gk=Rν\{0\}=G_0\subset\dots\subset G_k=\mathbb{R}^\nu and its rational exponents σ1<⋯<σk\sigma_1<\dots<\sigma_k attached to the system by Theorem 5 of the source, so that a direction yy lying in the stratum Gj∖Gj−1G_j\setminus G_{j-1} carries exponent σj=ρ(y)\sigma_j=\rho(y) — the analogue, for general hypoelliptic constant-coefficient systems, of the classical mixed estimates for the heat equation. The one-direction case must reproduce the published estimate ∣⟨y,D⟩μu(x)∣≤Cμ+1Γ(ρ(y)μ)\lvert\langle y,D\rangle^\mu u(x)\rvert \le C^{\mu+1}\Gamma(\rho(y)\mu) (Theorem 3 of the source), and the heat equation, with its time exponent 22 and space exponent 11, must be recovered as the model special case. Include the sharpness counterpart in the sense of Theorem 4 of the source: any bound valid for all solutions must dominate the optimal mixed growth, up to the constants. Also treat the inhomogeneous system P(D)u=f\mathcal{P}(D)u=f with Gevrey-type derivative bounds assumed on ff — the extension the source attributes to a method of Schwartz in its second remark after Theorem 3.

(2) Extension beyond the hypoelliptic class. Extend the directional (and mixed) estimates to systems that are not hypoelliptic but for which the exponent ρ(y)\rho(y) exists — the extension announced in the source's first remark after Theorem 3: 'The validity of this theorem is not restricted to hypoelliptic systems only but can easily be extended with slight modifications to any case where the number ρ(y)\rho(y) exists. The present case seems, however, to be the most interesting one.' This requires formulating the correct hypotheses on the system and on the class of solutions — and proving the resulting estimates in that setting.

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