Weighted Sobolev inequality in Musielak–Orlicz space

Let α, β, p and q be all variable exponents. Our aim in this paper is to deal with weighted Sobolev inequality for Riesz potentials of order α with functions f in Musielak–Orlicz spaces Lp,q,β(Rn) of variable exponent. Here p(⋅) and q(⋅) are variable exponents satisfying the log-Hölder and the log–log-Hölder conditions, respectively.