Convergence over fractals for the Schrödinger equation

Abstract

We consider a fractal refinement of the Carleson problem for the Schrödinger equation, that is to identify the minimal regularity needed by the solutions to converge pointwise to their initial data almost everywhere with respect to the α-Hausdorff measure (α-a.e.). We extend to the fractal setting (α < n) a recent counterexample of Bourgain [5], which is sharp in the Lebesque measure setting (α = n). In doing so we recover the necessary condition from [23] for pointwise convergence α-a.e. and we extend it to the range n/2 < α ≤ (3n + 1)/4.

Counterexamples for the fractal Schrödinger convergence problem with an Intermediate Space Trick

“…Lucà and Rogers [21] proved this for (3n+1)/4 ≤ α ≤ n, for which they constructed counterexamples based on ergodic arguments, different from Bourgain's one in [4] that is based on number theoretic arguments. Lucà and the second author adapted Bourgain's example to the fractal setting in [22] to prove (1.3) in the whole range.…”

“…The counterexamples combine the fractal extension of Bourgain's counterexample as presented in [22], and the intermediate space trick of Du-Kim-Wang-Zhang [12]. In [10], Du exploited this trick to construct counterexamples for (1.2), which are morally equivalent to counterexamples for convergence, except for one essential thing: for convergence the weight w must intersect every line t → (x, t) in at most one interval of length 1.…”