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

Abstract

Abstract Let T be the singular integral operator with variable kernel defined by T f ( x ) = p . v . ∫ R n Ω ( x , x − y ) | x − y | n f ( y ) d y $$\begin{array}{} \displaystyle Tf(x)= p.v. \int\limits_{\mathbb{R}^{n}}\frac{{\it\Omega}(x,x-y)}{|x-y|^{n}}f(y)\text{d}y \end{array} $$ 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 exponents H M K ˙ p ( ⋅ ) , λ α ( ⋅ ) , q $\begin{array}{} HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda} \end{array} $ via the convolution operator T m, j and Calderón-Zygmund operator, and then establish their boundedness on these spaces. The boundedness on H M K ˙ p ( ⋅ ) , λ α ( ⋅ ) , q $\begin{array}{} HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda} \end{array} $ (ℝ n ) is shown to hold for TDγ − DγT and (T ∗ − T ♯)Dγ . Moreover, the authors also establish various norm characterizations for the product T 1 T 2 and the pseudo-product T 1 ∘ T 2.