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Method of rotations for bilinear singular integrals

Commun. Math. Anal. Conf.
Conference 3 (2011), 99 - 107

Method of rotations for bilinear singular integrals

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Abstract

Suppose that W lies in the Hardy space H1 of the unit circle S1 in R2. We use the Calder´on–Zygmund method of rotations and the uniform boundedness of the bilinear Hilbert transforms to show that the bilinear singular operator with the rough kernel p.v.W(x=jxj)jxj¡2 is bounded from Lp(R)£Lq(R) to Lr(R), for a large set of indices satisfying 1=p+1=q = 1=r. We also provide an example of a function W in Lq(S1) with mean value zero to show that the singular integral operator given by convolution with p.v.W(x=jxj)jxj¡2 is not bounded from Lp1 (R)£Lp2 (R) to Lp(R) for 1=2 < p < 1, 1 < p1; p2 < ¥, 1=p1 +1=p2 = 1=p, 1 · q < ¥, and 1=p+1=q > 2: