Pointwise convergence over fractals for dispersive equations with homogeneous symbol

Abstract

We study the problem of pointwise convergence for equations of the type i ∂tu + P (D)u = 0, where the symbol P is real, homogeneous and non-singular. We prove that for initial data f ∈ H s (R n ) with s > (n − α + 1)/2 the solution u converges to f H α -a.e, where H α is the αdimensional Hausdorff measure. We improve upon this result depending on the dispersive strength of the symbol. On the other hand, we prove negative results for a large family of polynomial symbols P . Given α, we exploit a Talbot-like effect to construct regular initial data whose solutions u diverge in sets of Hausdorff dimension α. We also construct data for quadratic symbols like the saddle to show that our positive results are sometimes best possible. To compute the dimension of the sets of divergence we use a Mass Transference Principle from Diophantine approximation theory.

Decoupling for finite type phases in higher dimensions

“…For the study of necessary conditions , we refer to [1,12]. Z. Li, J. Zhao and T. Zhao [19] give a sufficient condition for the almost everywhere convergence problem associated with this degenerate phase in R 2 via the methods in [11].…”