\(L^2\)-boundedness of Hilbert transforms along variable curves (Q448243)
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scientific article; zbMATH DE number 6074402
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | \(L^2\)-boundedness of Hilbert transforms along variable curves |
scientific article; zbMATH DE number 6074402 |
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\(L^2\)-boundedness of Hilbert transforms along variable curves (English)
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30 August 2012
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Hilbert transform
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variable curves
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For \(f\in S(\mathbb{R}^2)\) and \(\Theta: \mathbb{R}^3\to\mathbb{R}^2\), a Hilbert transform along a curve is defined by NEWLINENEWLINE\[NEWLINEH_\Phi(f)(x)= \text{p.v. }\int^\infty_{-\infty} f(x-\Phi(x,t))\,{dt\over t}.NEWLINE\]NEWLINE NEWLINEThis kinds of operators have been extensively studied by A. Nagel, J. Vance, S. Wainger, D. Weinberg, J. M. Bennett etc. In this paper the authors first give a counterexample for \(L^2\)-boundedness of \(H_\Phi\) in special cases of \(\Phi(x, t)\). Simultaneously, \textit{J. M. Bennett}'s result [Trans. Am. Math. Soc. 354, No. 12, 4871--4892 (2002; Zbl 1017.44003)] is improved.
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