Sharp pointwise convergence of Schrödinger mean with complex time in higher dimensions

TL;DR

This work analyzes almost everywhere convergence of Schrödinger means with complex time $P_{\gamma}$ in higher dimensions, establishing a sharp Sobolev regularity threshold $s_0 = \min\left\{ \frac{d}{2(d+1)}, \frac{d}{d+1}\left(1-\frac{1}{\gamma}\right)^{+} \right\}$ for $d\ge 2$ and $\gamma>0$. It proves a matching $L^{2}$ maximal-function bound for frequency-localized data using a Littlewood–Paley framework and a torus Fourier-expansion argument, thereby obtaining almost everywhere convergence for $s>s_0$. The necessity part, built via a Nikišin–Stein type construction, shows the bound cannot be improved and identifies the regime-dependent contributors (Abel summation, Weyl sums, Gauss sums, Vitali). Together, the results quantify how complex time dampening with $e^{-t^{\gamma}| abla|^{2}}$ interacts with dimension to relax regularity requirements and extend Carleson-type convergence to dissipative complex-time Schrödinger means.

Abstract

In this paper, we establish the almost everywhere convergence of solutions to the Schrödinger operator with complex time $ P_γf(x,t) $ in higher dimensions, under the assumption that the initial data $f$ belongs to the Sobolev space $ H^{s}(\mathbb{R}^d)$.

Theorems & Definitions (12)

• Theorem 1.1
• Remark
• Proposition 2.1
• Lemma 2.2
• proof
• Remark
• Proposition 3.1
• Lemma 3.2: Continuous Abel summation
• Lemma 3.3: Quadratic Weyl sum estimate
• Lemma 3.4: Gauss sum evaluation