Bilinear Hilbert Transforms and (Sub)Bilinear Maximal Functions along Convex Curves

Abstract

In this paper, we determine the L p (R) × L q (R) → L r (R) boundedness of the bilinear Hilbert transform H γ ( f, g) along a convex curve γwhere p, q, and r satisfy 1 p + 1 q = 1 r , and r > 1 2 , p > 1, and q > 1. Moreover, the same L p (R) × L q (R) → L r (R) boundedness property holds for the corresponding (sub)bilinear maximal function

$L^p(\mathbb{R}^2)$-boundedness of Hilbert Transforms and Maximal Functions along Plane Curves with Two-variable Coefficients

“…We give some remarks for the hypothesis above. (i) of (H.), including some other similar forms, has been applied in the study of the Hilbert transforms along the variable curves (t, u(x 1 )γ(t)) with one-variable coefficients (e.g., [30,13]), and the bilinear Hilbert transform along the curves (t, γ(t)) ( [29]). (ii) and (iii) of (H.) imply the following "doubling property" of the curves.…”