Littlewood-Paley theory for variable exponent Lebesgue spaces and Gagliardo-Nirenberg inequality for Riesz potentials

Abstract

Our aim in this paper is to prove the Gagliardo-Nirenberg inequality for Riesz potentials of functions in variable exponent Lebesgue spaces, which are called Musielak-Orlicz spaces with respect to $\Phi(x,t)=t^{p(x)}(\log(c_0+t))^{q(x)}$ for $t$ > 0 and $x \in {\mathbb R}^n$, via the Littlewood-Paley decomposition.