Electronic Journal of Differential Equations (Dec 2006)
A counterexample to an endpoint bilinear Strichartz inequality
Abstract
The endpoint Strichartz estimate $$ | e^{itDelta} f |_{L^2_t L^infty_x(mathbb{R} imes mathbb{R}^2)} lesssim |f|_{L^2_x(mathbb{R}^2)} $$ 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 $$ | (e^{itDelta} P f) (e^{itDelta} P' g) |_{L^1_t L^infty_x(mathbb{R} imes mathbb{R}^2)} lesssim |f|_{L^2_x(mathbb{R}^2)} |g|_{L^2_x(mathbb{R}^2)} $$ fails even when $P$, $P'$ have widely separated supports.
