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 $D^{\gamma}(0\leq\gamma\leq1)$ be the fractional differentiation operator, where $K(x,z)=\frac{\Omega(x,z')}{|z|^{n}}$, $z'=\frac{z}{|z|},~~z\neq0$. Let $~T^{\ast}~$and $~T^\sharp~$ 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 $TD^{\gamma}-D^{\gamma}T$ and $(T^{\ast}-T^{\sharp})D^{\gamma}$ 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.