On Tempered Ultradistributions in Classical Sobolev Spaces

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.