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 Tf(X) = p.v.integral(Rn)Omega(x,x-y)/|x-y|(n)f(y)dy and D-gamma(0 <= gamma <= 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 TD gamma - (DT)-T-gamma and (T* - T-#)D-gamma on the homogeneous Morrey-Herz-type Hardy spaces with variable exponents HM\l=K:\alpha(.),q (p(.),lambda)via the convolution operator T-m,T-j and Calderon-Zygmund operator, and then establish their boundedness on these spaces. The boundedness on HM\l=K:\alpha(.),q (p(.),lambda)(R-n) is shown to hold for TD gamma -(DT)-T-gamma and (T* - T-#)D-gamma. Moreover, the authors also establish various norm characterizations for the product T1T2 and the pseudo-product T-1 \sh=cir\ T-2.