Campanato–Morrey spaces for the double phase functionals with variable exponents

Abstract

Our aim in this paper is to show that the Riesz potential operator Iα(⋅) of variable order α(⋅) embeds from variable exponent Morrey spaces Lp(⋅),ν(⋅)(G) to Campanato–Morrey spaces in the case α(x)p(x)=ν(x). Our result extends the recent work of Rafeiro and Samko (2019) and the authors (Mizuta et al. 2020). We show that Iα(⋅) embeds from Morrey spaces LΦ,ν(⋅)(G) of the double phase functionals Φ(x,t)=tp(x)+(b(x)t)q(x) to Campanato–Morrey spaces, where p(⋅) and q(⋅) satisfy log-Hölder conditions, p(x)<q(x) and b(⋅) is nonnegative, bounded and Hölder continuous of order θ∈(0,1]. We also discuss the continuity of Riesz potentials Iα(⋅)f of functions in LΦ,ν(⋅)(G) and show that Iα(⋅) embeds from LΦ,ν(⋅)(G) to vanishing Campanato–Morrey spaces.