On Tempered Ultradistributions in Classical Sobolev Spaces
A. U. Amaonyeiro, M. E. Egwe
TL;DR
The paper investigates the integration of tempered ultradistributions into the classical Sobolev framework by constructing Sobolev-type spaces $\mathcal{W}_{\mathcal{U}}^{l,p}$ built from $\mathcal{U}'$ and characterized via $\hat{\mu}$ with the weight $(1+|\xi|^{p})^{l}$. It proves that these spaces possess a Hilbert (Banach) structure, establish their embedding relations with $\mathcal{S}$, $\mathcal{U}$, and $\mathcal{U}'$, and analyze the action of constant-coefficient ultradifferential operators within this setting. Key contributions include continuity results for ultradifferential operators, embedding theorems, a Rellich-type compactness extension for $\mathcal{W}_{\mathcal{U}}^{l,p}$, and a representation of elements of $\mathcal{W}_{\mathcal{U}}^{l,p}$ as sums of derivatives of $L^{2}$-functions. Overall, the work provides a unified framework linking tempered ultradistributions with classical Sobolev spaces, enabling refined analysis of PDEs in generalized ultradifferentiable contexts.
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.