Electron. J. Diff. Eqns., Vol. 2006(2006), No. 151, pp. 1-6.

A counterexample to an endpoint bilinear Strichartz inequality

Terence Tao

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.

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Terence Tao
Department of Mathematics
University of California
Los Angeles, CA 90095-1555, USA
email: tao@math.ucla.edu

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