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

Abstract

Let T be the singular integral operator with variable kernel defined by $Tf(x)=p.v.{\int }_{{\mathbb{R}}^n}K(x,x-y)f(y)\mathrm{d}y$ and $D^3(0\le {\hat{I}}^3\le 1)$ be the fractional differentiation operator, where $K(x,z)=\frac{\hat{I}\text{c◯}(x,z^{\prime })}{|z{|}^n}$, $z^{\prime }=\frac{z}{|z|},z\ne 0$. Let $T^{\ast }$and $T^{♯}$ 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 $T{D}^3-D^{3}T$ and $(T^{\ast }-T^{♯})D^3$ 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.