Global hypoellipticity and solvability for a class of evolution operators in time-periodic weighted Sobolev spaces

TL;DR

This paper investigates the hypoellipticity and solvability of certain time-periodic evolution operators with polynomially growing coefficients, using specially designed weighted Sobolev spaces characterized by Fourier expansions.

Contribution

It introduces a new class of time-periodic weighted Sobolev spaces tailored for analyzing evolution operators with polynomial growth coefficients.

Findings

Established conditions for hypoellipticity of the operators.

Proved solvability results in the new weighted Sobolev spaces.

Connected Fourier analysis with elliptic operator properties.

Abstract

We study the hypoellipticity and solvability properties of a class of time-periodic evolution operators, with coefficients globally defined on $\mathbb{R}^d$ and growing polynomially with respect to the space variable. To this aim, we introduce a class of time-periodic weighted Sobolev spaces, whose elements are characterised in terms of suitable Fourier expansions, associated with elliptic operators.