Isomorphism theorems for some parabolic initial-boundary value problems in H\"ormander spaces

TL;DR

This paper establishes isomorphism theorems for second-order parabolic PDEs with boundary conditions in H"ormander spaces, providing a refined regularity framework beyond Sobolev spaces.

Contribution

It proves that the operators for these parabolic problems are isomorphisms in H"ormander spaces, characterizing regularity with a pair of parameters and a function parameter.

Findings

Operators are isomorphisms between H"ormander spaces.

H"ormander spaces describe regularity more finely than Sobolev spaces.

Regularity characterized by parameters and a Karamata-type function.

Abstract

In H\"ormander inner product spaces, we investigate initial-boundary value problems for an arbitrary second order parabolic partial differential equation and the Dirichlet or a general first-order boundary conditions. We prove that the operators corresponding to these problems are isomorphisms between appropriate H\"ormander spaces. The regularity of the functions which form these spaces is characterized by a pair of number parameters and a function parameter varying regularly at infinity in the sense of Karamata. Owing to this function parameter, the H\"ormander spaces describe the regularity of functions more finely than the anisotropic Sobolev spaces.