The bilinear Hilbert transform is the bilinear singular integral operator
 |
initially defined for Schwartz functions, where 
denotes the Cauchy principal value. Unlike
the Hilbert transform, it multiplies two translated
inputs before integration (Lacey and Thiele 1999).
Lacey and Thiele (1999) proved boundedness from 
to 
when 
, 
, and 
. Hu (2026) reports the endpoint bound
 |
for 
with 
. The output is the weak
L2/3 space, defined by a distribution-function bound, rather
than the strong L2/3-space.
Bounds for the Bilinear Hilbert Transform." 10
Nov 2026.