Weighted inequalities for fractional Hardy operators and commutators

Wenming Li ,Dong Liu,Jing Liu

doi:10.1186/s13660-019-2108-5

Wenming Li , Dong Liu + Show 1 more

Open Access

https://doi.org/10.1186/s13660-019-2108-5

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Abstract

In this paper, we introduce a fractional maximal operators N_{alpha } on (0,infty ) associated to the fractional Hardy operator P_{alpha } and its dual Q_{alpha }, 0leq alpha <1, and obtain some characterizations for the one-weight and two-weight inequalities for N_{alpha }. We also give some A_{p} type sufficient conditions for the two-weight inequalities for the fractional Hardy operators, the dual operators and the commutators of the fractional Hardy operators with CMO functions.

Highlights

Let Pα and Qα be the fractional Hardy operator and its adjoint on (0, ∞), 1x Pαf (x) = x1–α 0 f (y) dy, ∞ f (y) Qαf (x) =x y1–α dy, where 0 ≤ α < 1
The fractional Calderón operator Sα is defined as Sα = Pα + Qα
When α = 0, S0 is denoted S, and S is the Calderón operator, which plays a significant role in the theory of real interpolation; see [1]

Summary

Introduction

Let Pα and Qα be the fractional Hardy operator and its adjoint on (0, ∞), 1x Pαf (x) = x1–α 0 f (y) dy,. The fractional Calderón operator Sα is defined as Sα = Pα + Qα. We introduce the fractional maximal operators Nα related to. Li et al Journal of Inequalities and Applications (2019) 2019:158 the fractional Calderón operator. Given a measurable function f on (0, ∞), the fractional maximal operator Nα is defined as. Duoandikoetxea, Martin-Reyes and Ombrosi in [3] introduced the maximal operator N related to the Calderón operator and studied the weighted inequalities for N. Let b be a locally integrable function on (0, ∞), we define the commutators of the fractional Calderón operator Sα with b as Sαb = Pαb + Qbα, where

Pαb f

But for not bounded from

Thus we obtain

Thus we have s