On the dimension of divergence sets of dispersive equations

Abstract

We refine results of Carleson, Sjögren and Sjölin regarding the pointwise convergence to the initial data of solutions to the Schrödinger equation. We bound the Hausdorff dimension of the sets on which convergence fails. For example, with initial data in H 1 (R 3 ), the sets of divergence have dimension at most one.

A note on pointwise convergence for the Schrödinger equation

“…The bound is best possible in one dimension, in which case the central intervals are empty and the dimension jumps by a half over s = 1/4. This is a consequence of the Dahlberg-Kenig example combined with [1], where it was proven that α n (s) ≤ n − 2s in the range n/4 ≤ s ≤ n/2. For the best known upper bounds with lower regularity, see [14,Theorem 1.2].…”

“…In order to understand the behaviour of e i t 2πR ∆ f θ we first consider the unmodulated version e i t 2πR ∆ g. Barceló, Bennett, Carbery, Ruiz and Vilela [2] showed that 1 We write a b (a b) whenever a and b are nonnegative quantities that satisfy a ≤ Cb (a ≥ Cb) for a constant C > 0. We write a ≃ b when a b and b a.…”

Pointwise Convergence of the Schrödinger Flow

“…Indeed, recalling (99) and summing over N , this would imply the validity of (94), (95), (96) for this term. We could have required a weaker bound than (102), replacing the X 0+, 1 2 + δ norm with an X α+σ, 1 2 + δ norm if J 2 = II and with a (− ln ε) factor if J 2 = I. However, we are able to prove the stronger estimate (102).…”

“…In this work, we are concerned with the question of almost everywhere convergence of solutions to the nonlinear Schrödinger equation (NLS) to initial data. More precisely, let u(x, t) be a solution to (1) i∂ t u + ∆u = N (u), u(x, 0) = f (x),…”

“…pointwise convergence to the initial data. In order to obtain a good enough bound, in Proposition 3.3 we embed the restriction space X s, 1 2 + δ into the space…”