Weighted bounds for a class of singular integral operators in variable exponent Herz-Morrey spaces

Let T be the singular integral operator with variable kernel defined by $Tf(x)=p\mathrm{.}v\mathrm{.}{\int }_{{\mathbb{R}}^n}K(x,x-y)f(y)\mathrm{d}y$ and $D^{\gamma }(0\le \gamma \le 1)$ be the fractional differentiation operator, where $K(x,z)=\frac{\mathrm{\Omega }(x,z^{\mathrm{\prime }})}{\mathrm{\mid }z{\mathrm{\mid }}^n}$, $z^{\mathrm{\prime }}=\frac{z}{\mathrm{\mid }z\mathrm{\mid }},z\mathrm{\ne }0$. Let $T^{\ast }$and $T^{\mathrm{♯}}$ be the adjoint of $T$ and the pseudo-adjoint of $T$, respectively. In this paper, via the expansion of spherical harmonics and the estimates of the convolution operators $T_{m,j}$, we shall prove some boundedness results for $T{D}^{\gamma }-D^{\gamma }T$ and $(T^{\ast }-T^{\mathrm{♯}})D^{\gamma }$ under natural regularity assumptions on the exponent function on a class of generalized Herz-Morrey spaces with weight and variable exponent, which extend some known results. Moreover, various norm characterizations for the product $T_1{T}_2$ and the pseudo-product $T_1\circ T_2$ are also established.