On the maximal directional Hilbert transform

Abstract

For any dimension $n\ge 2$, we consider the maximal directional Hilbert transform ${\mathcal{H}}_U$ on ${\mathbb{R}}^n$ associated with a direction set $U\subseteq {\mathbb{S}}^{n-1}$: \[ \mathscr{H}_Uf(x) := \frac{1}π \sup_{v \in U} \Bigl| \text{p.v.} \int f(x - tv) \, \frac{dt}{t}\Bigr|.\] 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 $n\ge 2$ and any $p\in (1,\infty )$. This completes a result of Karagulyan, who proved a similar statement for $n=2$ and $p=2$.

29 pages, 8 figures. Minor revisions and updates

Cited by in corpus (1)

• On the maximal directional Hilbert transform in three dimensions