On pointwise convergence of cone multipliers

TL;DR

This paper proves the pointwise convergence of cone multipliers for functions with Fourier support in a specific cone, using weighted estimates and trace inequalities, extending understanding of Fourier multiplier behavior.

Contribution

It establishes new pointwise convergence results for cone multipliers under certain conditions, employing weighted maximal operator estimates and trace inequalities.

Findings

Pointwise convergence of cone multipliers for p ≥ 2.

Conditions on λ for convergence based on p and dimension.

Use of weighted estimates and trace inequalities as main tools.

Abstract

For $p\ge 2$, and $\lambda>\max\{n|\tfrac 1p-\tfrac 12|-\tfrac12, 0\}$, we prove the pointwise convergence of cone multipliers, i.e. $$ \lim_{t\to\infty}T_t^\lambda(f)\to f \text{ a.e.},$$ where $f\in L^p(\mathbb R^n)$ satisfies $supp\ \widehat f\subset\{\xi\in\mathbb R^n:\ 1<|\xi_n|<2\}$. Our main tools are weighted estimates for maximal cone operators, which are consequences of trace inequalities for cones.