Twisted Hilbert transforms vs Kakeya sets of directions

Abstract

Abstract Given a discrete group G ${\mathrm {G}}$ and an orthogonal action γ : G → O ( n ) ${\gamma : \mathrm {G}\rightarrow {\rm O}(n)}$ we study Lp -convergence of Fourier integrals which are frequency supported on the semidirect product ℝ n ⋊ γ G ${\mathbb {R}^n \rtimes _\gamma \mathrm {G}}$ . Given a unit u ∈ ℝ n ${u \in \mathbb {R}^n}$ and 1 < p ≠ 2 < ∞ ${1 < p \ne 2 < \infty }$ , our main result shows that the twisted (directional) Hilbert transform H u ⋊ γ id G ${H_u \rtimes _\gamma {\rm id}_\mathrm {G}}$ is Lp -bounded iff the orbit 𝒪 γ ( u ) ${\mathcal {O}_\gamma (u)}$ is finite. This is in sharp contrast with twisted Riesz transforms R u ⋊ γ id G ${R_u \rtimes _\gamma {\rm id}_\mathrm {G}}$ , which are always bounded. Our result characterizes Fourier summability in Lp 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.