×

Functional dimension of solution space of differential operators of constant strength. (English) Zbl 1447.35124

Summary: A differential operator with constant coefficients is hypoelliptic if and only if its solution space is of finite functional dimension. We extend this property to operators with variable coefficient. We prove that an equally strong differential operator with variable coefficients has the same property. In addition, we extend the Zielezny’s result to operators with variable coefficients; prove that an operator with analytic coefficients on \(\mathbb{R}^{\mathrm{n}}\) is elliptic if and only if locally the functional dimension of its solution space is the same as the Euclidean dimension \(n\).

MSC:

35H10 Hypoelliptic equations
35S35 Topological aspects for pseudodifferential operators in context of PDEs: intersection cohomology, stratified sets, etc.
47G30 Pseudodifferential operators
PDFBibTeX XMLCite
Full Text: Link