Estimates of Intrinsic Square Functions on Generalized Weighted Morrey Spaces

Guilian Gao,Xiaomei Wu

doi:10.1155/2014/168381

Guilian Gao, Xiaomei Wu

Open Access

https://doi.org/10.1155/2014/168381

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Abstract

We prove the boundedness of the intrinsic functions on generalized weighted Morrey spacesMp,φ(w), including the strong type estimates and weak type estimates. Moreover, we define thekth-order commutators generated byBMORnfunctions and intrinsic functions, and obtain their strong type estimates onMp,φ(w). In some cases, we improve previous results.

Highlights

We prove the boundedness of the intrinsic functions on generalized weighted Morrey spaces Mp,φ(w), including the strong type estimates and weak type estimates
It is well known that the theory of Littlewood-Paley square functions plays an important role in harmonic analysis, such as in the study of Fourier multiplier and singular integral operators
Wang [12, 16] studied the boundedness of the intrinsic functions and their commutators on weighted Morrey spaces

Summary

Introduction

It is well known that the theory of Littlewood-Paley square functions plays an important role in harmonic analysis, such as in the study of Fourier multiplier and singular integral operators. ≤ Cφ2 (x, r) , where C does not depend on x and r Under these conditions, the boundedness of some classical operators and commutators in generalized weighted Morrey spaces were obtained, respectively. Wang [12, 16] studied the boundedness of the intrinsic functions and their commutators on weighted Morrey spaces. In [6, 17], the authors obtained the boundedness of the intrinsic functions and their commutators on generalized Morrey spaces. Guliyev et al [7,8,9] obtained the estimates for vector-valued intrinsic square functions and their kthorder commutators on vector-valued generalized weighted Morrey spaces. We prove the boundedness of the intrinsic functions and their kth-order commutators on generalized weighted Morrey spaces Mp,φ(w) under the conditions (11), (12), and w ∈ Ap, respectively.

Definitions and Main Results

Lemmas

Proofs of Main Theorems