A Remark on reverse Littlewood-Paley, restriction and Kakeya

TL;DR

This paper explores how a conjectured optimal reverse Littlewood-Paley inequality could lead to sharp results in Kakeya maximal functions, Bochner-Riesz means, and Fourier restriction, linking these areas in harmonic analysis.

Contribution

It establishes a connection between a conjectured reverse Littlewood-Paley inequality and sharp bounds in key harmonic analysis problems.

Findings

Potential implications for Kakeya maximal function bounds

Implications for sharp Bochner-Riesz means results

Connections to Fourier restriction conjecture

Abstract

We show that a certain conjectured optimal reverse Littlewood- Paley inequality would, if true, imply sharp results for the Kakeya maximal function, the Bochner-Riesz means and the Fourier restriction operator.