Semilinear pseudodifferential equations in spaces of tempered ultradistributions

TL;DR

This paper investigates semilinear elliptic equations within spaces of tempered ultradistributions, establishing regularity results for solutions when the linear operator has sub-exponential growth and the nonlinearity involves infinite sums.

Contribution

It introduces a novel analysis of semilinear elliptic equations with infinite order pseudodifferential operators in ultradistribution spaces, proving regularity of weak solutions.

Findings

Regularity of solutions in ultradistribution spaces.

Handling of infinite order pseudodifferential operators.

Extension of elliptic regularity theory to ultradistributions.

Abstract

We study a class of semilinear elliptic equations on spaces of tempered ultradistributions of Beurling and Roumieu type. Assuming that the linear part of the equation is an elliptic pseudodifferential operator of infinite order with a sub-exponential growth of its symbol and that the non linear part is given by an infinite sum of powers of $u$ with sub-exponential growth with respect to $u,$ we prove a regularity result in the functional setting of the quoted ultradistribution spaces for a weak Sobolev type solution $u$.