## $$L^p$$ 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

### Keywords:

norms; $$L^p$$ estimates; bilinear Hilbert transform; inequality
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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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