On Tempered Ultradistributions in Classical Sobolev Spaces

TL;DR

This paper develops a new framework for tempered ultradistributions within Sobolev spaces, establishing their properties, embeddings, and extending classical theorems like Rellich's compactness theorem.

Contribution

It introduces a novel Sobolev space preserving ultradistribution properties and proves new embedding theorems and an extension of Rellich's theorem.

Findings

Defined a new Sobolev space with ultradistribution properties

Proved embedding theorems involving rapidly decreasing functions

Extended Rellich's compactness theorem to this setting

Abstract

We construct and investigate the properties of tempered ultradistribution spaces in Sobolev spaces. A new Sobolev space preserving the original properties and condition whose derivatives are linear continuous operators embedding in $L^p$ for $1\leq p\leq \infty$ is characterized. Moreover, we also consider some Sobolev embedding theorems involving rapidly decreasing functions, and finally, we prove the extension of Rellich's compactness theorem.