On a Muckenhoupt-Type Condition for Morrey Spaces

Abstract

As is known, the class of weights for Morrey type spaces \({{\mathcal{L}^{p,\lambda}(\mathbb{R}^n)}}\) for which the maximal and/or singular operators are bounded, is different from the known Muckenhoupt class A p of such weights for the Lebesgue spaces L p(Ω). For instance, in the case of power weights \({{|x - a|^{\nu}, a \in \mathbb{R}^1}}\) , \({a \in \mathbb {R}^1}\) , the singular operator (Hilbert transform) is bounded in \({{L^p(\mathbb{R})}}\) , if and only if −1 < ν < p − 1, while it is bounded in the Morrey space \({{\mathcal{L}^{p,\lambda}(\mathbb{R}), 0 \leq \lambda < 1}}\) , if and only if the exponent α runs the shifted interval λ − 1 < ν < λ + p − 1. A description of all the admissible weights similar to the Muckenhoupt class A p is an open problem. In this paper, for the one-dimensional case, we introduce the class A p,λ of weights, which turns into the Muckenhoupt class A p when λ = 0 and show that the belongness of a weight to A p,λ is necessary for the boundedness, in Morrey spaces, of the Hilbert transform in the one-dimensional case. In the case n > 1 we also provide some λ-dependent á priori assumptions on weights and give some estimates of weighted norms \({{\|\chi B\|_{p,\lambda ;w}}}\) of the characteristic functions of balls.