Sobolev's embedding on time scales.

Abstract

For $1\le p<n$ , the embeddings of Sobolev spaces $W_{\Delta }^{1,p}\left({\Omega }_{T^n}\right)$ of functions defined on an open subset of an arbitrary time scale $T^n$ , $n\ge 1$ , endowed with the Lebesgue Δ-measure have been developed in (Agarwal et al. in Adv. Differ. Equ. 2006:38121, 2006) for $n=1$ and later generalized to arbitrary $n\ge 1$ in (Su et al. in Dyn. Partial Differ. Equ. 12(3):241-263, 2015). In this article we present the embeddings of Sobolev spaces $W_{\Delta }^{1,p}\left({\Omega }_{T^n}\right)$ for $n\le p\le \infty$ and then, using these embeddings, we develop general Sobolev's embedding for the Sobolev spaces $W_{\Delta }^{1,p}\left({\Omega }_{T^n}\right)$ on time scales, where k is a non-negative integer and $1\le p\le \infty$ .