It is shown that the bilinear Hilbert transforms H α,β(f,g)(x) = p.v. ∫R f(x - αt)g(x - βt) dt/t map Lp1(R) × Lp2(R) → L p2(R) uniformly in the real parameters α,β when 2 < p1,p2 < ∞ and 1 < p = p1p 2/p1p2 < 2. Combining this result with the main result in , we deduce that the operators H1,α map L2(R) × L∞(R) → L2(R) uniformly in the real parameter α ∈ [0, 1]. This completes a program initiated by A. Calderón.
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty