On the functional dimension of solution spaces of hypoelliptic partial differential operators
Let P(D) denote a hypoelliptic pdo with constant coefficients and \(N_ P:=\{f\in C^{\infty}({\mathbb{R}}^ n)| \quad P(D)f=0\}.\) If P is equivalent to \(d_ P^{\deg P}\) for \(dp(\zeta):=\inf \{| z-\zeta | | \quad P(z)=0\}\) and if \(| \zeta_ n| \leq C(d_ P(\zeta)+1),\) then the functional dimension df \(N_ P\) of \(N_ P\) may be calculated: \[ (*)\quad df N_ P=1+\deg P\lim_{t\to \infty}(\ln v(t)/\ln t) \] for \(v(t):=\lambda \{x\in {\mathbb{R}}^{n-1}| | P(x,0)| \leq t\}\), \(\lambda=\) Lebesgue-measure. (*) is used to estimate df \(N_ H\) for general hypoelliptic polynomials H, improving classical results of \textit{Y. Kōmura} [Funkc. Ekvacioj, Ser. Int. 9, 313-324 (1966; Zbl 0185.396)]. The proof is entirely different from the reasoning of that paper: \(N_ P\) is represented as the dual of a weighted space of entire functions to estimate df \(N_ P\). (*) implies that the estimate df \(N_ P>n\) (= number of variables) for nonelliptic P given by \textit{Z. Zielezny} [J. Differ. Equations 18, 340-345 (1975; Zbl 0307.35018)] is incorrect. The right side of (*) however provides an estimate from below for df \(N_ P\), if P is hypoelliptic and deg P\(=\deg_{\zeta_ n}P.\) This implies the following characterization of elliptic operators (weaker than the claim of Z. Zielezny): P is elliptic iff P is equivalent to \(d_ P^{\deg P}\) and df \(N_ P=n\).
scientific article; zbMATH DE number 3886386