Weighted Morrey spaces of variable exponent and Riesz potentials

Abstract

We prove weighted inequalities for the Hardy-Littlewood maximal operator on weighted Morrey spaces L p(·),ν (R n ; μ) of variable exponent. As an application of the boundedness of the maximal operator, we establish weighted Sobolev's inequality for Riesz potentials. We are also concerned with weighted Trudinger's inequality for Riesz potentials.

Extrapolation to weighted Morrey spaces with variable exponents and applications

“…In this paper, we further extend the studies in [20, 42] to establish the mapping properties of pseudo-differential operators and Fourier integral operators on weighted Morrey spaces with variable exponents. We obtain our main results by extending the well-known extrapolation theory introduced by Rubio de Francia [51–52] to the weighted Morrey spaces with variable exponents.…”

“…For the mapping properties of the singular integral operators and the Riesz potentials on the weighted Morrey spaces with variable exponents, the reader is referred to [20, 42].…”

“…The weighted Morrey spaces with variable exponents were introduced and studied in [20, 42]. Moreover, the mapping properties of singular integral operators and the Riesz potentials were obtained in [19] and [42], respectively.…”

Boundedness of fractional integral operators in Herz spaces on the hyperplane

“…Capone, Cruz‐Uribe and Fiorenza 15 studied the boundedness of the fractional maximal operator M α in . See Cruz‐Uribe et al 16 For double phase functionals, we refer to Mizuta et al 17,18 See Mizuta and Shimomura 19 for weighted Morrey spaces of variable exponents, Xu and Yang 20 for variable exponent Herz‐type Besov and Triebel‐Lizorkin spaces and Mizuta et al 21 for Herz‐Morrey‐Orlicz spaces on the half space.…”

Variable exponent weighted norm inequality for generalized Riesz potentials

“…These results have been extended to the case of variable exponent; see [1] and [4] for unweighted spaces, and [11], [12] and [14] for weighted spaces.…”