The bilinear Hilbert transform is pointwise finite

Abstract

Let $f \in L^{\infty}$ and $g \in L^{2}$ be supported on [0,1]. Then the principal value integral below exists in $L^{1}$ .

p.v. \int f (x + y) g (x - y) \frac{d y}{y} .

Michael T. Lacey, The bilinear Hilbert transform is pointwise finite. Rev. Mat. Iberoam. 13 (1997), no. 2, pp. 411–469