ESTIMATES FOR BILINEAR Θ-TYPE CALDERÓN-ZYGMUND OPERATORS AND THEIR COMMUTATORS ON NON-HOMOGENEOUS GENERALIZED WEIGHTED MORREY SPACES

Let $ (\mathcal{X}, d, \mu) $ be a non-homogeneous metric measure space satisfying geometrically doubling and upper doubling conditions. Under assumption that a dominating function $ \lambda $ satisfies $ \varepsilon $-weak reverse doubling condition, the authors prove that a bilinear $ \theta $-type Calderón-Zygmund operator $ \widetilde{T}_{\theta} $ is bounded from product of generalized weighted Morrey spaces $ \mathcal{L}^{p_{1}, \Phi, \varrho}_{\omega_{1}}(\mu) \times \mathcal{L}^{p_{2}, \Phi, \varrho}_{\omega_{2}}(\mu) $ into weak generalized weighted Morrey spaces $ W\mathcal{L}^{p, \Phi, \varrho}_{\nu_{\vec{\omega}}}(\mu) $, and also show that the commutator $ \widetilde{T}_{\theta, b_{1}, b_{2}} $ generated by $ b_{1}, b_{2}\in\widetilde{\mathrm{RBMO}}(\mu) $ and $ \widetilde{T}_{\theta} $ are bounded from product of spaces $ \mathcal{L}^{p_{1}, \Phi, \varrho}_{\omega_{1}}(\mu) \times \mathcal{L}^{p_{2}, \Phi, \varrho}_{\omega_{2}}(\mu) $ into spaces $ W\mathcal{L}^{p, \Phi, \varrho}_{\nu_{\vec{\omega}}}(\mu) $, where $ \Phi: (0, \infty)\rightarrow (0, \infty) $ is a Lebesgue measurable function, $ \varrho\in(1, \infty) $, $ \vec{p}=(p_{1}, p_{2}) $, $ \vec{\omega}=(\omega_{1}, \omega_{2})\in A^{\tau}_{\vec{p}}(\mu) $, $ \nu_{\vec{\omega}}\in RH_{r}(\mu) $ for $ r\in(1, \infty) $, and $ \frac{1}{p}=\frac{1}{p_{1}}+ $ $ \frac{1}{p_{2}} $ with $ 1<p_{1}, p_{2}<\infty $. Furthermore, the strong and weak type results for the $ \widetilde{T}_{\theta} $ and $ \widetilde{T}_{\theta, b_{1}, b_{2}} $ on the product of spaces $ \mathcal{L}^{p_{1}, \Phi, \varrho}_{\omega_{1}}(\mu) \times \mathcal{L}^{p_{2}, \Phi, \varrho}_{\omega_{2}}(\mu) $ are established.