HK-Sobolev space $W{S}^{k,p}$ on ${\mathbb{R}}^{\infty }$ and Bessel Potential

Abstract

Our goal in this article is to construct HK-Sobolev spaces on ${\mathbb{R}}^{\infty }$ which contains Sobolev spaces as dense embedding. We discuss that the sequence of weak solution of Sobolev spaces are convergence strongly in HK-Sobolev space. Also, we obtain that the Sobolev space through Bessel Potential is densely contained in HK-Sobolev spaces. Finally we find sufficient condition for the solvability of the divergence equation $\nabla .F=f,$ for $f$ is an element of the subspace $K{S}^p[{\mathbb{R}}_I^n]$ and $n\in \mathbb{N}$, in the SoboHK-Sobolev space $W{S}^{k,p}[{\mathbb{R}}_I^n]$ with the help of Fourier transformation.

no of pages 28. We welcome the comments and suggestions from the expert in this area