Geodesic
A counterexample to an endpoint bilinear Strichartz inequality
Electronic journal of differential equations, Tome 2006 (2006)
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

$ \| (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.
Classification : 35J10
Keywords: Strichartz inequality 
@article{EJDE_2006__2006__a52,
     author = {Tao,  Terence},
     title = {A counterexample to an endpoint bilinear {Strichartz} inequality},
     journal = {Electronic journal of differential equations},
     year = {2006},
     volume = {2006},
     zbl = {1128.35315},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a52/}
}
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Tao,  Terence. A counterexample to an endpoint bilinear Strichartz inequality. Electronic journal of differential equations, Tome 2006 (2006). http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a52/