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

摘要

Our aim in this paper is to show that the Riesz potential operator I-alpha(.) of variable order alpha(.) embeds from variable exponent Morrey spaces L-p(.),nu((.))(G) to Campanato-Morrey spaces in the case alpha(x)p(x) = nu(x). Our result extends the recent work of Rafeiro and Samko (2019) and the authors (Mizuta et al. 2020). We show that I-alpha(.) embeds from Morrey spaces L-Phi,L-nu(.)(G) of the double phase functionals Phi(x, t) - t(p(x))+(b(x)t)(q(x)) to Campanato-Morrey spaces, where p(.) and q(.) satisfy log-Holder conditions, p(x) < q(x) and b(.) is nonnegative, bounded and Holder continuous of order theta is an element of (0, 1]. We also discuss the continuity of Riesz potentials I-alpha(.) f of functions in L-Phi,L-nu(.)(G) and show that I-alpha(.) embeds from LF,.(center dot)(G) to vanishing Campanato-Morrey spaces. (C) 2020 Elsevier Ltd. All rights reserved.