Two-Weight Norm Inequality for the One-Sided Hardy-Littlewood Maximal Operators in Variable Lebesgue Spaces

Caiyin Niu,Zongguang Liu,Panwang Wang

doi:10.1155/2016/1648281

Abstract

The authors establish the two-weight norm inequalities for the one-sided Hardy-Littlewood maximal operators in variable Lebesgue spaces. As application, they obtain the two-weight norm inequalities of variable Riemann-Liouville operator and variable Weyl operator in variable Lebesgue spaces on bounded intervals.

Highlights

Introduction and Main ResultsThe variable Lebesgue space Lp(⋅)(E) is the set of measurable functions f on E such that ρp(⋅),E (f) fl ∫ 󵄨󵄨󵄨󵄨f (x)󵄨󵄨󵄨󵄨p(x) dx < ∞
Introduction and Main ResultsThe one-sided Hardy-Littlewood maximal operators M+ and M− are defined by M+f (x) = sup h>0 1 h x+h ∫ x 󵄨󵄨󵄨󵄨f (t)󵄨󵄨󵄨󵄨 dt, (1) M−f
Throughout this paper, u and V are nonnegative locally integrable functions and C is a positive constant whose value may change from one occurrence to the

Summary

Introduction and Main Results

The variable Lebesgue space Lp(⋅)(E) is the set of measurable functions f on E such that ρp(⋅),E (f) fl ∫ 󵄨󵄨󵄨󵄨f (x)󵄨󵄨󵄨󵄨p(x) dx < ∞. The weighted variable Lebesgue space Lpw(⋅)(E) is the set of measurable functions f on E such that fw ∈ Lp(⋅)(E) and ‖f‖Lpw(⋅)(E) = ‖fw‖Lp(⋅)(E). Throughout this paper, u and V are nonnegative locally integrable functions and C is a positive constant whose value may change from one occurrence to the next. Sp+(⋅)(E) and Sp󸀠+(⋅)(E) can be considered to be the generalization of Sawyer-type two-weight condition (see [1]) for the one-sided maximal operator in variable exponents case. If we change the conditions Sp+(⋅), Sp󸀠+(⋅), P+σ to Sp−(⋅), Sp󸀠−(⋅), and P−σ, respectively, in above theorems, we will obtain similar results of M−. Corresponding results for variable Lebesgue spaces can be found in [13,14,15]

Proof of the Main Results

Applications