Singular Radon transforms along odd curves on the Heisenberg group (Q816227)

Let \(\mathbb{H}^1\) be the three-dimensional Heisenberg group identified with \(\mathbb{R}^2\times \mathbb{R}\), and let \(\Gamma (t)=(t, \gamma (t),0)\) be a curve living on the hyperplane of the Heisenberg group \(\mathbb{H}^1\). Set \(h(t)=t\gamma' (t)-\gamma (t)\). Suppose \(h\) is infinitesimally doubling, namely, there exists a constant \(\varepsilon >0\) such that \(h'(t)>\varepsilon h(t)/t\) for all \(t>0\). Define a maximal function along the curve \(\Gamma(t)\) by \[ M_{\Gamma}f(x)=\sup_{r>0}\frac{1}{r}\int_0^r\left| f\left(x\cdot \Gamma(t)^{-1}\right)\right| dt, \] where \(\gamma\) is extended as an odd or even function in \(\mathbb{R}^1\), and define the Hilbert transform along the curve \(\Gamma\) by \[ H_{\Gamma}f(x)=\text{ p.\, v.}\,\int_{-\infty}^{\infty}f\left(x\cdot \Gamma(t)^{-1}\right)\frac{dt}{t}. \] In this paper, the author proves that the maximal operator \(M_{\Gamma}\) is bounded on \(L^p(\mathbb{H}^1)\) with \(p \in (1, \infty ]\), and if \(\gamma\) is extended to be an odd function, then \(H_{\Gamma}\) is bounded on \(L^p(\mathbb{H}^1)\) with \(p \in (1, \infty)\). The proof is based on the generalized Calderón-Zygmund theory on the generalized space of homogeneous type.