On the maximal directional Hilbert transform

TL;DR

This paper proves that the maximal directional Hilbert transform's operator norm grows at least as the square root of the logarithm of the number of directions, implying unboundedness for infinite directions in any dimension.

Contribution

It establishes a lower bound on the operator norm of the maximal directional Hilbert transform for finite sets of directions in any dimension, extending previous results to all p in (1, ∞).

Findings

Operator norm grows at least as C_{p,n} * sqrt(log #U)

Infinite direction sets lead to unbounded transforms on L^p

Completes a conjecture for all dimensions and p in (1, ∞)

Abstract

For any dimension $n \geq 2$, we consider the maximal directional Hilbert transform $\mathscr{H}_U$ on $\mathbb R^n$ associated with a direction set $U \subseteq \mathbb S^{n-1}$: \[ \mathscr{H}_Uf(x) := \frac{1}{\pi} \sup_{v \in U} \Bigl| \text{p.v.} \int f(x - tv) \, \frac{dt}{t}\Bigr|.\] The main result in this article asserts that for any exponent $p \in (1, \infty)$, there exists a positive constant $C_{p,n}$ such that for any finite direction set $U \subseteq \mathbb S^{n-1}$, \[||\mathscr{H}_U||_{p \rightarrow p} \geq C_{p,n} \sqrt{\log \#U}, \] where $\#U$ denotes the cardinality of $U$. As a consequence, the maximal directional Hilbert transform associated with an infinite set of directions cannot be bounded on $L^p(\mathbb{R}^{n})$ for any $n\geq 2$ and any $p \in (1, \infty)$. This completes a result of Karagulyan, who proved a similar statement for $n=2$ and $p=2$.