On a Muckenhoupt-type condition for Morrey spaces

Abstract

As is known, the class of weights for Morrey type spaces $\mathcal{L}^{p,\lb}(\rn) $ 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 $L^p(\Om)$. For instance, in the case of power weights $|x-a|^\nu, \ a\in \mathbb{R}^1,$ the singular operator (Hilbert transform) is bounded in $L^p(\mathbb{R})$, if and only if $-1<\nu <p-1$, while it is bounded in the Morrey space $\mathcal{L}^{p,\lb}(\mathbb{R}), 0\le \lb<1$, if and only if the exponent $\al$ runs the shifted interval $\lb-1<\nu <\lb+p-1.$ 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 $A_{p,\lb}$ of weights, which turns into the Muckenhoupt class $A_p$ when $\lb=0$ and show that the belongness of a weight to $A_{p,\lb}$ is necessary for the boundedness of the Hilbert transform in the one-dimensional case. In the case $n>1$ we also provide some $\lb$-dependent \textit{\`a priori} assumptions on weights and give some estimates of weighted norms $\|\chi_B\|_{p,\lb;w}$ of the characteristic functions of balls.