A counterexample to an endpoint bilinear Strichartz inequality

Abstract

The endpoint Strichartz estimate $\Vert e^{it\Delta }f{\Vert }_{L_t^2{L}_x^{\infty }(\mathbb{R}\times {\mathbb{R}}^2)}\lesssim \Vert f{\Vert }_{L_x^2({\mathbb{R}}^2)}$ is known to be false by the work of Montgomery-Smith, despite being only ``logarithmically far'' from being true in some sense. In this short note we show that (in sharp constrast to the $L_{t,x}^p$ Strichartz estimates) the situation is not improved by passing to a bilinear setting; more precisely, if $P,P^{\prime }$ are non-trivial smooth Fourier cutoff multipliers then we show that the bilinear estimate $$\Vert (e^{it\Delta }Pf)(e^{it\Delta }{P}^{\prime }g){\Vert }_{L_t^2{L}_x^{\infty }(\mathbb{R}\times {\mathbb{R}}^2)}\lesssim \Vert f{\Vert }_{L_x^2({\mathbb{R}}^2)}\Vert g{\Vert }_{L_x^2({\mathbb{R}}^2)}$$ fails even when $P$, $P^{\prime }$ have widely separated supports.

7 pages, no figures, submitted, EJDE