Annals of Functional Analysis

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

Zheng Wang, Weiliang Xiao, and Taotao Zheng

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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 $\mathbb{R}^{n}$.

Article information

Source
Ann. Funct. Anal., Volume 6, Number 4 (2015), 134-154.

Dates
First available in Project Euclid: 1 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.afa/1435764007

Digital Object Identifier
doi:10.15352/afa/06-4-134

Mathematical Reviews number (MathSciNet)
MR3365987

Zentralblatt MATH identifier
1321.42039

Subjects
Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)

Keywords
Bilinear Calderón--Zygmund operator sharp maximal operator non-homogeneous space upper doubling measures weight

Citation

Zheng, Taotao; Wang, Zheng; Xiao, Weiliang. Maximal bilinear Calderón--Zygmund operators of type $\omega(t)$ on non-homogeneous space. Ann. Funct. Anal. 6 (2015), no. 4, 134--154. doi:10.15352/afa/06-4-134. https://projecteuclid.org/euclid.afa/1435764007


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