On a Muckenhoupt-type condition for Morrey spaces

TL;DR

This paper introduces a new weight class $A_{p,eta}$ for Morrey spaces, extending Muckenhoupt weights, and establishes necessary conditions for the boundedness of the Hilbert transform in these spaces.

Contribution

It defines the class $A_{p,eta}$ for Morrey spaces, generalizing Muckenhoupt weights, and proves its necessity for Hilbert transform boundedness in one dimension.

Findings

The class $A_{p,eta}$ reduces to $A_p$ when $eta=0$.

Necessary conditions for Hilbert transform boundedness in Morrey spaces are established.

Provides estimates for weighted norms of characteristic functions of balls.

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.