Lacunary directional maximal function on the Heisenberg group (Q1769975)

Let \(K=(k_1,k_2,k_3)\in{\mathbb Z}^3\) and \(v_K=(2^{k_1}, 2^{k_2}, 2^{k_3}).\) Let \(\mathbb H^1\) be the Heisenberg group identified with \({\mathbb R}^2\times {\mathbb R}^1\) endowed with the group multiplication: \[ (p,q,t)\cdot(p',q',t')=(p+p',q+q', t+t'+2(p'\cdot q-p\cdot q')). \] For any \(f\in L^1_{\text{ loc}}(\mathbb H^1)\), define \[ {\mathcal M}^{\text{ lac}}f(x)=\sup_{K\in {\mathbb Z}^3,\, r>0} \frac 1{2r}\int^r_{-r}f(x\cdot(tv_K)^{-1})\,dt. \] The author proves that for \(1<p\leq\infty\), \({\mathcal M}^{\text{lac}}\) is a bounded operator on \(L^p(\mathbb H^1)\) with an operator norm depending on \(p\). The proof is based on the group Fourier transform and the angular Littlewood-Paley decompositions in the Heisenberg group by the author in [Duke Math. J. 114, No. 3, 555--593 (2002; Zbl 1012.42013)].