Sobolev embeddings for Riesz potential spaces of variable exponents near 1 and Sobolev's exponent

Abstract

Our aim in this paper is to deal with Sobolev embeddings for Riesz potentials of order α for functions f satisfying the Orlicz type condition ∫ | f ( y ) | p ( y ) ( log ( c + | f ( y ) | ) ) q ( y ) d y < ∞ , where p ( ⋅ ) and q ( ⋅ ) are variable exponents satisfying the log-Hölder conditions.