An extension of the Muckenhoupt–Wheeden theorem to generalized weighted Morrey spaces

Abstract

Abstract In this paper, we find the condition on a function ω and a weight v which ensures the equivalency of norms of the Riesz potential and the fractional maximal function in generalized weighted Morrey spaces ℳ p , ω ⁢ ( ℝ n , v ) {{\mathcal{M}}_{p,\omega}({\mathbb{R}}^{n},v)} and generalized weighted central Morrey spaces ℳ ˙ p , ω ⁢ ( ℝ n , v ) {\dot{\mathcal{M}}_{p,\omega}({\mathbb{R}}^{n},v)} , when v belongs to the Muckenhoupt A ∞ {A_{\infty}} -class.