Maximal bilinear Calderón--Zygmund operators of type $\omega(t)$ on non-homogeneous space

Abstract

Let $(\mathcal{X},d,\mu)$ be a geometrically doubling metric space and assume that the measure $\mu$ satisfies the upper doubling condition. In this paper, the authors, by invoking a Cotlar type inequality, show that the maximal bilinear Calderón--Zygmund operators of type $\omega(t)$ is bounded from $L^{p_{1}}(\mu)\times L^{p_{2}}(\mu)$ into $L^{p}(\mu)$ for any $p_{i}\in(1,\infty]$ and bounded from $L^{p_{1}}(\mu)\times L^{p_{2}}(\mu)$ into $L^{p,\infty}(\mu)$ for $p_{1}=1$ or $p_{2}=1$, where $p \in [1/2,\infty)$, $1/{p_{1}}+1/{p_{2}}=1/{p}$. Moreover, if $\vec{w}=(w_{1},w_{2})$ belongs to the weight class $A_{\vec{p}}^{\rho}(\mu)$, using the John-strömberg maximal operator and the John-strömberg sharp maximal operator, the authors obtain a weighted weak type estimate $L^{p_{1}}(w_{1}) \times L^{p_{1}}(w_{2}) \rightarrow L^{p,{\infty}}(v_{\vec{w}})$ for the maximal bilinear Calderón--Zygmund operators of type $\omega(t)$. By weakening the assumption of $\omega\in \mathrm{Dini}(1/2)$ into $\omega\in \mathrm{Dini}(1)$, the results obtained in this paper are substantial improvements and extensions of some known results, even on Euclidean spaces $\mathbb{R}^{n}$.