An isomorphism theorem for parabolic problems in H\"ormander spaces and its applications

TL;DR

This paper proves an isomorphism theorem for parabolic problems in anisotropic H"ormander spaces, enabling new insights into solution regularity and conditions for the continuity of derivatives.

Contribution

It establishes an isomorphism between operators and H"ormander spaces for parabolic problems, advancing the understanding of solution regularity in these spaces.

Findings

Operators are isomorphisms between H"ormander spaces for the problem.

Theorem on local increase in regularity of solutions.

New conditions for continuity of generalized derivatives.

Abstract

We investigate a general parabolic initial-boundary value problem with zero Cauchy data in some anisotropic H\"ormander inner product spaces. We prove that the operators corresponding to this problem are isomorphisms between appropriate H\"ormander spaces. As an application of this result, we establish a theorem on the local increase in regularity of solutions to the problem. We also obtain new sufficient conditions under which the generalized derivatives, of a given order, of the solutions should be continuous.