Dimension of divergence sets of oscillatory integrals with concave phase

TL;DR

This paper investigates the Hausdorff dimension of divergence sets for solutions to fractional Schrödinger equations with concave phases, revealing different behaviors from classical cases, especially in one-dimensional settings.

Contribution

It provides new insights into the divergence sets of fractional Schrödinger solutions with concave phases for m in (0,1), including their Hausdorff dimensions and convergence properties.

Findings

Hausdorff dimension of divergence sets characterized

Different divergence behaviors along curves and lines

Distinct from classical cases with m > 1

Abstract

We study the Hausdorff dimension of the sets on which the pointwise convergence of the solutions to the fractional Schr\"odinger equation $e^{it(-\Delta)^\frac m2}f$ fails when $m\in(0,1)$ in one spatial dimension. The pointwise convergence along a non-tangential curve and a set of lines are also considered, where we find different nature from the case when $m\in(1,\infty)$.