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\(L^p\) estimates for the bilinear Hilbert transform. (English) Zbl 0946.44001
Summary: For the bilinear Hilbert transform given by \[ Hfg(x)=\text{p.v. }\int f(x-y)g(x+y){dy\over y}, \] we announce the inequality \(\|H fg\|_{p_3}\leq K_{p_1,p_2}\|f\|_{p_1}\|g\|_{p_2}\), provided \(2<p_1, p_2<\infty\), \(1/p_3=1/p_1+1/p_2\) and \(1<p_3<2\).

MSC:
44A15 Special integral transforms (Legendre, Hilbert, etc.)
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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References:
[1] Calderon, PNAS 53 (5) pp 1092– (1965) · Zbl 0151.16901
[2] ACTA MATH 116 pp 135– (1966) · Zbl 0144.06402
[3] ANN MATH 98 pp 551– (1973) · Zbl 0268.42009
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