Boundedness of commutators of rough fractional Hardy operators on grand Herz-Morrey spaces with variable exponents

다국어 초록 (Multilingual Abstract)

In this paper, we will investigate the boundedness of the commutators of the fractional Hardy operator $\mathcal{H}_{\Omega,\beta(\cdot)}$ of variable order $\beta(\cdot)$ with some rough kernel $\Omega\in L^{s}(\mathbb{S}^{n-1}) (s>1)$ and its dual operator $\mathcal{H}^{*}_{\Omega,\beta(\cdot)}$ generated with the function $b\in \mathrm{CMO}^{q(\cdot)}(\mathbb{R}^{n})$ on the grand variable Herz-Morrey spaces $M\dot{K}^{\alpha(\cdot),u),\theta}_{\lambda,p(\cdot)}(\mathbb{R}^{n})$, respectively. It's worth noting that, compared with spaces $\mathrm{BMO}(\mathbb{R}^{n})$, the spaces $\mathrm{CMO}^{q(\cdot)}(\mathbb{R}^{n})$ are not equipped with the property $\mathrm{BMO}^{q(\cdot)}(\mathbb{R}^{n})\thickapprox\mathrm{BMO}(\mathbb{R}^{n})$, which brings some difficulties to establish main results. Moreover, the corresponding results are also new even on the grand variable Herz spaces $\dot{K}^{\alpha(\cdot),u),\theta}_{p(\cdot)}(\mathbb{R}^{n})$.