ON THE MAXIMAL DIRECTIONAL HILBERT TRANSFORM

For any dimension n >= 2, we consider the maximal directional Hilbert transform HU69pt}parallel to HU parallel to p -> p >= Cp,|_{p \to p}} \geq {C_{p,n}}\sqrt {\log \# U} $$\end{document}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(Double-struck capital R-n) for any n >= 2 and any p is an element of (1,infinity). This completes a result of Karagulyan [11], who proved a similar statement for n = 2 and p = 2.