Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 151, pp. 1-6.
Title: A counterexample to an endpoint bilinear Strichartz inequality
Author: Terence Tao (Univ. of California, Los Angeles, CA, USA)
Abstract:
The endpoint Strichartz estimate
$$
\| e^{it\Delta} f \|_{L^2_t L^\infty_x(\mathbb{R} \times \mathbb{R}^2)}
\lesssim \|f\|_{L^2_x(\mathbb{R}^2)}
$$
is known to be false by the work of Montgomery-Smith
[2], despite being only "logarithmically
far" from being true in some sense. In this short note we show
that (in sharp contrast to the $L^p_{t,x}$
Strichartz estimates) the situation is not improved by passing
to a bilinear setting; more precisely, if $P, P'$ are non-trivial
smooth Fourier cutoff multipliers then we show that
the bilinear estimate
$$
\| (e^{it\Delta} P f) (e^{it\Delta} P' g) \|_{L^1_t L^\infty_x(\mathbb{R} \times \mathbb{R}^2)}
\lesssim \|f\|_{L^2_x(\mathbb{R}^2)} \|g\|_{L^2_x(\mathbb{R}^2)}
$$
fails even when $P$, $P'$ have widely separated supports.
Submitted September 29, 2006. Published December 5, 2006.
Math Subject Classifications: 35J10.
Key Words: Strichartz inequality.