On the maximal directional Hilbert transform

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 $U \subseteq \mathbb S^{n-1}$: $${\mathcal{H}}_{U}f(x):=\frac{1}{\pi }\underset{v\in U}{\sup }\mid \text{p.v.}\int f(x-tv)\frac{dt}{t}\mid \mathrm{.}$$ The main result in this article asserts that for any exponent $p\in (1,\mathrm{\infty })$, there exists a positive constant $C_{p,n}$ such that for any finite direction set $U \subseteq \mathbb S^{n-1}$, $$\mathrm{\mid }\mathrm{\mid }{\mathcal{H}}_U\mathrm{\mid }{\mathrm{\mid }}_{p\to p}\ge C_{p,n}\sqrt{\log \mathrm{\#}U},$$ where $\mathrm{\#}U$ 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 $p\in (1,\mathrm{\infty })$. This completes a result of Karagulyan, who proved a similar statement for $n=2$ and $p=2$.