A Carleson problem for the Boussinesq operator

TL;DR

This paper investigates the pointwise convergence of the Boussinesq operator to initial data in Sobolev spaces, establishing the optimal regularity threshold and the Hausdorff dimension of the disconvergence set, with extensions to higher dimensions for radial functions.

Contribution

It determines the optimal Sobolev regularity for convergence and characterizes the disconvergence set's Hausdorff dimension, extending results to higher dimensions for radial data.

Findings

Optimal convergence index s=1/4 established.

Hausdorff dimension of disconvergence set varies with s.

Higher dimensional results for radial functions provided.

Abstract

In this paper, Theorems 1.1- 1.2 show that the Boussinesq operator $\mathcal{B}_tf$ converges pointwise to its initial data $f\in H^s(\mathbb{R})$ as $t\to 0$ provided $s\geq\frac{1}{4}$ -- more precisely -- on the one hand, by constructing a counterexample in $\mathbb{R}$ we discover that the optimal convergence index $s_{c,1}=\frac14$; on the other hand, we find that the Hausdorff dimension of the disconvergence set for $\mathcal{B}_tf$ is \begin{align*} \alpha_{1,\mathcal{B}}(s)&=\begin{cases} 1-2s&\ \ \text{as}\ \ \frac{1}{4}\leq s\leq\frac{1}{2};\\ 1 &\ \ \text{as}\ \ 0<s<\frac{1}{4}. \end{cases} \end{align*} Moreover, Theorem 1.3 presents a higher dimensional lift of Theorems 1.1- 1.2 under $f$ being radial.