| Volume 14, Number 6, 2024, Pages - DOI：10.11948/JAAC-2024-0026 |
| Estimates for bilinear $\Theta$-type Calder\ |
| Miaomiao Wang,Guanghui Lu,Shuangping Tao |
| Keywords:Non-homogeneous metric measure space bilinear $\theta$-type Calder\ |
| Abstract: |
| 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 $\lambda$ satisfi-\\es $\varepsilon$-weak reverse doubling condition, the authors prove that a bilinear $\theta$-type Calder\"{o}n-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
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ wi-\\th $1 |
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