On Tempered Ultradistributions in Classical Morrey Spaces

TL;DR

This paper introduces tempered ultradistributions in classical Morrey spaces, explores their properties, and establishes embedding theorems, advancing the understanding of function spaces with ultradistribution elements.

Contribution

It defines tempered ultradistributions within Morrey spaces and proves new embedding results for these spaces involving ultradistributions.

Findings

Established embedding theorems for Morrey spaces with ultradistributions.

Characterized properties of tempered ultradistributions in Morrey spaces.

Extended classical Morrey space theory to include ultradistribution elements.

Abstract

We introduce the notion of tempered ultradistributions in classical Morrey spaces by preserving their respective properties. Moreover we investigate some embedding results within the scale of classical Morrey spaces (local Morrey space $\mathcal{L}^{p,\beta}(\mathbb{C}^{n},\mu)$ or global Morrey space $L^{p,\beta}(\mathbb{C}^{n})$ where the underlying functions $\mu$ satisfy the growth condition by polxnomials. Finally we present some results on the new classical Morrey spaces described by tempered ultradistributions satisfying both the vanishing and slow growth conditions in form of embedding theorems.