Sobolev’s embedding on time scales

Abstract

Abstract For 1≤p<n $1 \leq p < n $, the embeddings of Sobolev spaces WΔ1,p(ΩTn) $W^{1,p}_{\Delta }(\Omega_{\mathbb{T}^{n}})$ of functions defined on an open subset of an arbitrary time scale Tn $\mathbb{T}^{n}$, n≥1 $n\geq1$, endowed with the Lebesgue Δ-measure have been developed in (Agarwal et al. in Adv. Differ. Equ. 2006:38121, 2006) for n=1 $n=1$ and later generalized to arbitrary n≥1 $n \geq1$ in (Su et al. in Dyn. Partial Differ. Equ. 12(3):241–263, 2015). In this article we present the embeddings of Sobolev spaces WΔ1,p(ΩTn) $W^{1,p}_{\Delta}(\Omega_{\mathbb {T}^{n}})$ for n≤p≤∞ $n\leq p \leq\infty$ and then, using these embeddings, we develop general Sobolev’s embedding for the Sobolev spaces WΔ1,p(ΩTn) $W^{1,p}_{\Delta}(\Omega_{\mathbb{T}^{n}})$ on time scales, where k is a non-negative integer and 1≤p≤∞ $1\leq p \leq\infty$.

Keywords

• Sobolev spaces
• Time scales
• Embedding