| Citation: |
Guanghui Lu, Miaomiao Wang. BILINEAR $\Theta$-TYPE CALDERÓN-ZYGMUND OPERATOR AND ITS COMMUTATOR ON PRODUCT NON-HOMOGENEOUS GENERALIZED MORREY SPACES[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2922-2942. doi: 10.11948/20230054
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BILINEAR $\Theta$-TYPE CALDERÓN-ZYGMUND OPERATOR AND ITS COMMUTATOR ON PRODUCT NON-HOMOGENEOUS GENERALIZED MORREY SPACES
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Abstract
Let $ (\mathcal{X}, d, \mu) $ be a non-homogeneous metric measure space satisfying both the geometrically doubling and upper doubling conditions in the sense of Hytönen. In this setting, the authors first introduce generalized Morrey spaces $ \mathcal{L}^{p, \varphi, \kappa}(\mu) $ and generalized weak Morrey spaces $ W\mathcal{L}^{p, \varphi, \kappa}(\mu) $ for $ p\in[1, $ $ \infty) $; second, under assumption that the dominating function $ \lambda $ and $ (\varphi_{1}, \varphi_{2}, \varphi) $ satisfy certain conditions, the authors prove that bilinear $ \theta $-type Calderón-Zygmund operators $ \widetilde{T} $ are bounded from product of spaces $ \mathcal{L}^{p_{1}, \varphi_{1}, \kappa}(\mu)\times \mathcal{L}^{p_{2}, \varphi_{2}, \kappa}(\mu) $ into spaces $ \mathcal{L}^{p, \varphi, \kappa}(\mu) $, and also bounded from product of spaces $ \mathcal{L}^{p_{1}, \varphi_{1}, \kappa}(\mu)\times \mathcal{L}^{p_{2}, \varphi_{2}, \kappa}(\mu) $ into spaces $ W\mathcal{L}^{p, \varphi, \kappa}(\mu) $, where $ \frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}} $ for $ 1<p_{1}, p_{2}<\infty $; finally, the boundedness of the commutator $ \widetilde{T}_{b_{1}, b_{2}} $ formed by $ b_{1}, b_{2}\in\mathrm{RBMO}(\mu) $ and $ \widetilde{T} $ on spaces $ \mathcal{L}^{p, \varphi, \kappa}(\mu) $ and on spaces $ W\mathcal{L}^{p, \varphi, \kappa}(\mu) $ is obtained.
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References
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Guanghui Lu, Miaomiao Wang. BILINEAR $\Theta$-TYPE CALDERÓN-ZYGMUND OPERATOR AND ITS COMMUTATOR ON PRODUCT NON-HOMOGENEOUS GENERALIZED MORREY SPACES[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2922-2942. doi: 10.11948/20230054
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