A counterexample to an endpoint bilinear Strichartz inequality

Abstract

The endpoint Strichartz estimate ∥eit∆ƒ∥L2t L∞x(ℝxℝ2 ≲ ∥ƒ∥L2x(ℝ2) 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 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 ∥(eit∆Pƒ) (eit∆P′g∥ L1t L∞x (ℝxℝ2) ≲ ∥ƒ∥L2x(ℝ2)∥g∥L2x(ℝ2) fails even when P, P′ have widely separated supports.