A Counterexample to an Endpoint Bilinear Strichartz Inequality
نویسنده
چکیده
The endpoint Strichartz estimate ‖ef‖L2 t L x (R×R) . ‖f‖L2 x (R) 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 constrast to the Lpt,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 ‖(ePf)(eP ′g)‖L2 t L x (R×R) . ‖f‖L2x(R2)‖g‖L2x(R2) fails even when P , P ′ have widely separated supports.
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تاریخ انتشار 2008