  We study the inclusion of tempered ultradistributions (or functions of slow
growth) in the notion of classical Sobolev spaces. We investigate basically the
properties of tempered ultradistribution spaces in Sobolev spaces. Our 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.
