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

Let T be the singular integral operator with variable kernel defined by $$\begin{array}{} \displaystyle Tf(x)= p.v. \int\limits_{\mathbb{R}^{n}}\frac{{\it\Omega}(x,x-y)}{|x-y|^{n}}f(y)\text{d}y \end{array} $$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 $\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 on $\begin{array}{} HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda} \end{array} $(ℝn) is shown to hold for TDγ − DγT and (T∗ − T♯)Dγ. Moreover, the authors also establish various norm characterizations for the product T1T2 and the pseudo-product T1 ∘ T2.