Commutators of linear and bilinear Hilbert transforms期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Commutators of linear and bilinear Hilbert transforms

Authors:	Oscar Blasco Paco Villarroya

Institution:	Departamento de Análisis Matemático, Universidad de Valencia, 46100 Burjassot, Valencia, Spain ; Departamento de Análisis Matemático, Universidad de Valencia, 46100 Burjassot, Valencia, Spain

Abstract:	Let $\alpha \in \mathbb{R}$ , and let $H_\alpha(f,g)(x)=\frac{1}{\pi} p.v. \int f(x-t)g(x-\alpha t)\frac{dt}{t}$ and $Hf(x)= \frac{1}{\pi} p.v.\int f(x-t)\frac{dt}{t}$ denote the bilinear and linear Hilbert transforms, respectively. It is proved that, for $1<p<\infty$ and $\alpha_1\ne\alpha_2$ , $H_{\alpha_1}-H_{\alpha_2}$ maps $L^p\times BMO$ into $L^{p}$ and it maps $BMO \times L^p$ into $L^{p}$ if and only if ${sign}(\alpha_1)={sign}(\alpha_2)$ . It is also shown that, for $\alpha\le1$ the commutator $H_{\alpha,f},H]$ is bounded on for $1<p<\infty$ if and only if $f\in BMO$ , where $H_{\alpha,f}(g)=H_\alpha(f,g)$ .

Keywords:	Bilinear Hilbert transform commutators

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