# Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents

Singular integrals with variable kernel and fractional differentiation in homogeneous... AbstractLet T be the singular integral operator with variable kernel defined byTf(x)=p.v.∫RnΩ(x,x−y)|x−y|nf(y)dy$$\begin{array}{}\displaystyle Tf(x)= p.v. \int\limits_{\mathbb{R}^{n}}\frac{{\it\Omega}(x,x-y)}{|x-y|^{n}}f(y)\text{d}y\end{array}$$and Dγ(0 ≤ γ ≤ 1) be the fractional differentiation operator. Let T∗ and T♯ be the adjoint of T and the pseudo-adjoint of T, respectively. The aim of this paper is to establish some boundedness for TDγ − DγT and (T∗ − T♯)Dγ on the homogeneous Morrey-Herz-type Hardy spaces with variable exponentsHMK˙p(⋅),λα(⋅),q$\begin{array}{}HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\end{array}$ via the convolution operator Tm, j and Calderón-Zygmund operator, and then establish their boundedness on these spaces. The boundedness onHMK˙p(⋅),λα(⋅),q$\begin{array}{}HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\end{array}$(ℝn) is shown to hold forTDγ − DγT and (T∗ − T♯)Dγ. Moreover, the authors also establish various norm characterizations for the product T1T2 and the pseudo-product T1 ∘ T2. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

# Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents

, Volume 16 (1): 20 – Apr 10, 2018
20 pages

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Publisher
De Gruyter
Copyright
© 2018 Yang and Tao, published by De Gruyter
ISSN
2391-5455
eISSN
2391-5455
D.O.I.
10.1515/math-2018-0036
Publisher site
See Article on Publisher Site

### Abstract

AbstractLet T be the singular integral operator with variable kernel defined byTf(x)=p.v.∫RnΩ(x,x−y)|x−y|nf(y)dy$$\begin{array}{}\displaystyle Tf(x)= p.v. \int\limits_{\mathbb{R}^{n}}\frac{{\it\Omega}(x,x-y)}{|x-y|^{n}}f(y)\text{d}y\end{array}$$and Dγ(0 ≤ γ ≤ 1) be the fractional differentiation operator. Let T∗ and T♯ be the adjoint of T and the pseudo-adjoint of T, respectively. The aim of this paper is to establish some boundedness for TDγ − DγT and (T∗ − T♯)Dγ on the homogeneous Morrey-Herz-type Hardy spaces with variable exponentsHMK˙p(⋅),λα(⋅),q$\begin{array}{}HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\end{array}$ via the convolution operator Tm, j and Calderón-Zygmund operator, and then establish their boundedness on these spaces. The boundedness onHMK˙p(⋅),λα(⋅),q$\begin{array}{}HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\end{array}$(ℝn) is shown to hold forTDγ − DγT and (T∗ − T♯)Dγ. Moreover, the authors also establish various norm characterizations for the product T1T2 and the pseudo-product T1 ∘ T2.

### Journal

Open Mathematicsde Gruyter

Published: Apr 10, 2018

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