  In this paper, we first prove some local estimates for bilinear operators
(closely related to the bilinear Hilbert transform and similar singular
operators) with truncated symbol. Such estimates, in accordance with the
Heisenberg uncertainty principle correspond to a description of
``off-diagonal'' decay. In addition they allow us to prove global continuities
in Lebesgue spaces for bilinear operators with spatial dependent symbol.

Functional Analysis;Classical Analysis and ODEs