MAPPING PROPERTY FOR BILINEAR $\Theta$-TYPE GENERALIZED FRACTIONAL INTEGRAL OPERATOR ON GRAND VARIABLE EXPONENTS HERZ-MORREY SPACES

The aim of this paper is to establish the boundedness of the bilinear $ \theta $-type generalized fractional integral operators $ B\widetilde{T}_{\beta,\theta} $ and their commutators $ B\widetilde{T}_{\beta,\theta,b_1,b_2} $, generated by $ b_1,b_2\in\mathrm{BMO}(\mathbb{R}^n) $ and $ B\widetilde{T}_{\beta,\theta} $, on Lebesgue spaces with variable exponent $ L^{q(\cdot)}(\mathbb{R}^{n}) $. Under the assumption that variable exponents $ \alpha(\cdot) $ and $ q_{i}(\cdot) $ for $ i=1,2 $ satisfy the $ \log $ decay at both infinity and the origin, the authors prove that the $ B\widetilde{T}_{\beta,\theta} $ and $ B\widetilde{T}_{\beta,\theta,b_1,b_2} $ are bounded on the grand variable exponents Herz spaces $ \dot{K}_{q(\cdot)}^{\alpha(\cdot),p),\theta}(\mathbb{R}^n)\big(\hbox{or}\ K_{q(\cdot)}^{\alpha(\cdot),p),\theta}(\mathbb{R}^n)\big) $ and grand variable exponents Herz-Morrey spaces $ M\dot{K}_{p),\theta,q(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^n)\big(\hbox{or}\ MK_{p),\theta,q(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^n)\big) $, respectively.