Sobolev-type theorem for commutators of variable Marcinkiewicz fractional in grand Herz-Morrey spaces

Abstract

Let $\mathbb{S}^{n-1}$ denote unit sphere in $\mathbb{R}^n$ equipped with the normalized Lebesgue measure. Let $\Phi \in L^r(\mathbb{S}^{n-1})$ be a homogeneous function of degree zero. The variable Marcinkiewicz fractional
integral operator is defined as
$$[b,\mu _{\Phi}]^m_\zeta (f)(z_1)= \left( \int \limits _0 ^ \infty \left|\int \limits _{|z_1-z_2| \leq s} \frac{\Phi(z_1-z_2)[b(z_1)-b(z_2)]^m}{|z_1-z_2|^{n-1-\zeta(z_1)}}f(z_2)dz_2\right|^2 \frac{ds}{s^3}\right)^{\frac{1}{2}}.$$

The commutators Marcinkiewicz fractional operator of variable order $\zeta(z_1)$ is shown to be bounded
from the grand Herz-Morrey spaces ${M\dot{K} ^{a(\cdot), u),\theta}_{\beta, p(\cdot)}(\mathbb{R}^n)}$ with variable exponent $p(z)$ into the
weighted space ${M\dot{K} ^{a(\cdot), u),\theta}_{\beta, \rho, q(\cdot)}(\mathbb{R}^n)}$, where $\rho=(1+|z_1|)^{-\lambda}$ and
${1 \over q(z)}={1 \over p(z)}-{\zeta(z_1) \over n}$when $p(z_1)$ is not necessarily constant at infinity.