Weak Estimates of Singular Integrals with Variable Kernel and Fractional Differentiation on Morrey-Herz Spaces

Yanqi Yang,Shuangping Tao

doi:10.1155/2017/4340805

Abstract

Let T be the singular integral operator with variable kernel defined by Tf(x)=p.v.∫Rn(Ω(x,x-y)/x-yn)f(y)dy and let Dγ (0≤γ≤1) be the fractional differentiation operator. Let T⁎and T♯ be the adjoint of T and the pseudoadjoint of T, respectively. In this paper, the authors prove that TDγ-DγT and (T⁎-T♯)Dγ are bounded, respectively, from Morrey-Herz spaces MK˙p,1α,λ(Rn) to the weak Morrey-Herz spaces WMK˙p,1α,λ(Rn) by using the spherical harmonic decomposition. Furthermore, several norm inequalities for the product T1T2 and the pseudoproduct T1∘T2 are also given.

Highlights

Denote Sn−1 to be the unit sphere in Rn (n ≥ 2) with normalized Lebesgue measure dσ
Let Iγ be the Riesz potential operator of order γ defined on the space of tempered distributions modulo polynomials by setting Î γf(ξ) = |ξ|−γf(ξ)
Let T1 and T2 be the operators defined in (1) which are differentiated by their kernels Ω1(x, y) and Ω2(x, y)

Summary

Introduction

The singular integral operator with variable kernel is defined by. Let Iγ be the Riesz potential operator of order γ defined on the space of tempered distributions modulo polynomials by setting Î γf(ξ) = |ξ|−γf(ξ). Unless otherwise stated, for a μ-measurable set E, χE denotes its characteristic function. Let T1 and T2 be the operators defined in (1) which are differentiated by their kernels Ω1(x, y) and Ω2(x, y). In 2016, Tao and Yang obtained the boundedness of those operators on the weighted Morrey-Herz spaces (see [6]). Assume that T is defined by (1) and Ω(x, y), which satisfies (2) and (3), meets the following condition: max. D is the square root of Laplacian operator and D0 is the identity operator I In this case, we obtain the following results. 󵄩󵄩󵄩󵄩(T1 ∘ T2 − T1T2) Df󵄩󵄩󵄩󵄩WMKpα,,1λ(Rn) ≲ 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩MKpα,,1λ(Rn)

Preliminaries and Main Lemmas

Proofs of Theorems