Geodesic
A counterexample to an endpoint bilinear Strichartz inequality
Electronic Journal of Differential Equations, Tome 2006 (2006).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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},
     publisher = {mathdoc},
     volume = {2006},
     year = {2006},
     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/