Sobolev Inequalities for Functions on Graphs

Abstract The aim of this paper is to study the Sobolev inequalities when 0 < p < 1. Sobolev spaces defined by the Russian mathematician Sorgi Sobolev. The need for these spaces that some results are found in these spaces rather than the spaces of continuous functions. Sobolev and some authors proved Sobolev inequality in the cases p= 1 and then p > 1. Here we prove Sobolev types inequalities in the case 0 < p < 1. An infinite dimensional Sobolev inequality for expander is also proved. Also we introduce estimates for Banach-Mazur distance between the spaces Sp(G) and Z- using Cheeger constant.