HK-Sobolev spaces $W{S^{k,p}}$ and Bessel Potential

Abstract

Our goal in this article is to construct HK-Sobolev spaces on $\R^\infty$ which contains Sobolev spaces as dense embedding. We show that weakly convergent sequences in Sobolev spaces are strongly convergent in HK-Sobolev spaces. Also, we obtain that the Sobolev space through Bessel potential is densely contained in HK-Sobolev spaces. Finally we find sufficient conditions for the solvability of the divergence equation $\nabla\cdot F= f,$ when $f$ is an element of the subspace $K{S^p}[\R_I^n]$ of the HK-Sobolev space $WS^{k,p}[\R_I^n] $ with the help of the Fourier transformation.