Sobolev type spaces associated with the Poly-axially operator

Abstract

In this paper n-dimensional Sobolev type spaces $E_{\alpha }^{s,p}({\mathbb{R}}_{+}^n)$ $(\alpha \in {\mathbb{R}}^n,{\alpha }_1>-\frac{1}{2},...,{\alpha }_n>-\frac{1}{2},s\in \mathbb{R},p\in [1,+\infty ])$ are defined on ${\mathbb{R}}_{+}^n$ by using Fourier-Bessel transform. Some properties including completeness and embedding results for these spaces are obtained, Poincaré's inequality and Reillich theorem are proved.

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