Riesz means associated with homogeneous functions on Hardy spaces (Q543029)

Suppose that \(\rho\in C^\infty(\mathbb R^n\setminus\{0\})\) is a homogeneous function of degree one. For a Schwartz function \(f\in \Im(\mathbb R^n)\), denote by \(\hat{f}(\xi)=\int_{\mathbb R^n} f(x)e^{-i\langle x,\xi\rangle}\,dx\) the Fourier transform of \(f\). The Riesz means \(S^\delta_k\) is defined by \[ \widehat{S^\delta_k f}(\xi)= \bigg(1-\frac{\rho(\xi)}{2^k} \bigg)^\delta_+ \hat{f}(\xi), \quad \xi\in\mathbb R^n, \] and the corresponding lacunary maximal function \[ S^\delta_*f(x)= \sup_{k\in\mathbb Z} \big|S^\delta_k f(x)\big|. \] The author proves a sharp weak type \((p,p)\) estimate on \(H^p(\mathbb R^n)\) \((1<p\leq 1)\) of the maximal lacunary Riesz operator \(S^\delta_*\) at the critical index \(\delta_p=n(1/p-1/2)-1/2\). Precisely, suppose that \(0<p\leq 1\) and \(\delta=\delta_p\), then \(S^\delta_*\) maps \(H^p(\mathbb R^n)\) boundedly into weak-\(L^p(\mathbb R^n)\). The proof is based on the Littlewood-Paley square-functions for \(H^p(\mathbb R^n)\) \((0<p\leq 1)\) and some ideas of \textit{M. Christ} and \textit{C. D. Sogge} [Invent. Math. 94, No. 2, 421--453 (1988; Zbl 0678.35096)]. This result implies the almost everywhere convergence of the operator \(S^\delta_kf\) on \(H^p(\mathbb R^n)\) \((0<p\leq 1)\) at the critical index.