Twisted Hilbert transforms vs Kakeya sets of directions
  Given a discrete group $\G$ and an orthogonal action $\gamma: \G \to O(n)$ we study $L_p$ convergence of Fourier integrals which are frequency supported on the semidirect product $\R^n \rtimes_\gamma \G$. Given a unit $u \in \R^n$ and $1 < p \neq 2 < \infty$, our main result shows that the twisted (directional) Hilbert transform $H_u \rtimes_\gamma id_\G$ is $L_p$-bounded iff the orbit $\mathcal{O}_\gamma(u)$ is finite. This is in sharp contrast with twisted Riesz transforms $R_u \rtimes_\gamma id_\G$, which are always bounded. Our result characterizes Fourier summability in $L_p$ for this class of groups. We also extend de Leeuw's compactification theorem to this setting and obtain stronger estimates for functions with "lacunary" frequency support. 
Javier Parcet and Keith M. Rogers
math.OA math.CA math.FA