Two Weight Characterization of New Maximal Operators

Hu Yunpeng,Cao Yonghui

doi:10.11648/j.pamj.20190803.11

Abstract

For the last twenty years, there has been a great deal of interest in the theory of two weight. In the present paper, we investigate the two weight norm inequalities for fractional new maximal operator on the Lebesgue space. More specifically, we obtain that the sufficient and necessary conditions for strong and weak type two weight norm inequalities for a new fractional maximal operators by introducing a class of new two weight functions. In the discussion of strong type two weight norm inequalities, we make full use of the properties of dyadic cubes and truncation operators, and utilize the space decomposition technique which space is decomposed into disjoint unions. In contrast, weak type two weight norm inequalities are more complex. We have the aid of some good properties of Ap weight functions and ingeniously use the characteristic function. What should be stressed is that the new two weight functions we introduced contains the classical two weights and our results generalize known results before. In this paper, it is worth noting that w(x)dx may not be a doubling measure if our new weight functions ω∈Ap (φ). Since φ(|Q|)≥1, our new weight functions are including the classical Muckenhoupt weights.

Highlights

In 1972, Muckenhopt [1] established the weight theory when studying the Lebesgue boundedness of classicalHardy-Littlewood maximal operators
Muckenhopt and Wheeden [2] founded that the two weight condition is only a necessary but not sufficient condition for Hardy-Littlewood maximal operator and Hilbert transform to have two weight strong boundedness, which is essentially different from the one weight case
As a generalization of the one weight case, it is more difficult to discuss the boundedness of operators with two weights than with one weight

Summary

Introduction

In 1972, Muckenhopt [1] established the weight theory when studying the Lebesgue boundedness of classicalHardy-Littlewood maximal operators. Muckenhopt and Wheeden [2] founded that the two weight condition is only a necessary but not sufficient condition for Hardy-Littlewood maximal operator and Hilbert transform to have two weight strong boundedness, which is essentially different from the one weight case. Be a doubling measure, i.e. there exist a constant > 0 for any cube such that The new maximal operators were firstly introduced by Tang

Objectives

Results

Conclusion