Caltech/UCLA Joint Analysis Seminar
The bilinear Hilbert transform is a prototypical modulation invariant multi-linear singular operator
BHT α (f 1 ,f 2 )(x):=∫ R ∫ R f 1 ˆ (ξ 1 )f 2 ˆ (ξ 2 )1 [0,+∞) (ξ 1 −αξ 2 )e 2πi(ξ 1 +ξ 2 )x dξ 1 dξ 2 . BHTα(f1,f2)(x):=∫R∫Rf1^(ξ1)f2^(ξ2)1[0,+∞)(ξ1−αξ2)e2πi(ξ1+ξ2)xdξ1dξ2.
It arises in many contexts including Cauchy integrals along Lipschitz curves and in the study of Calderón commutators.BHT satisfies the bounds
∥BHT α (f 1 ,f 2 )(x)∥ L p ′ 3 (R) ≤C p 1 ,p 2 ,α ∥f 1 ∥ L p 1 (R) ∥f 2 ∥ L p 2 (R) ‖BHTα(f1,f2)(x)‖Lp3′(R)≤Cp1,p2,α‖f1‖Lp1(R)‖f2‖Lp2(R)
for any p 1,2,3 ∈(1,+∞) p1,2,3∈(1,+∞) with 1p 1 +1p 2 +1p 3 =1 1p1+1p2+1p3=1 .A longstanding open problem is that of bounds for the bilinear Hilbert transform uniform in the parameter α α . The dyadic analog of this problem has been solved by Oberlin and Thiele (2010).We will give a general overview of the topic and of the main ideas involved in the proof of the above bound with a constant C p 1 ,p 2 Cp1,p2 independent of α α . We will particularly emphasize the role of the framework of outer measure L p Lp spaces of Do and Thiele in proving the result.