Riesz means associated with convex polygons in \(\mathbb R^2\)
Let \(\mathcal P\) be a convex polygon in \(\mathbb R^2\) containing the origin in its interior. Define a distance function \(\rho\) by \(\rho(\xi)=\inf\{t; t\xi\notin \mathcal P\}\), and a Bochner-Riesz like operator \(T_\varepsilon^\delta\) by \(\widehat{T_\varepsilon^\delta f}(\xi)=(1-(\rho(\xi)/\varepsilon)^2)_+^\delta \hat f(\xi)\), and its maximal operator \(T_*^\delta\) by \(T_*^\delta f(x) =\sup_{\varepsilon>0}| T_*^\delta f(x)| \). The authors' main results are: (1) If \(1/2<p<1\) and \(\delta=2(1/p-1)\) or \(p=1/2\) and \(\delta >2(1/p-1)\), then \(T_*^\delta\) maps \(H^p(\mathbb R^2)\) boundedly into weak-\(L^p(\mathbb R^2)\). (2) \(1/2<p<1\) and \(\delta >2(1/p-1)\) if and only if \(T_*^\delta\) maps \(H^p(\mathbb R^2)\) boundedly into \(L^p(\mathbb R^2)\). When \(\mathcal P\) is the unit circle, \(T_\varepsilon^\delta\) is the usual Bochner-Riesz mean of \(f\), and studied by \textit{E. M. Stein, M. H. Taibleson} and \textit{G. Weiss} [Suppl. Rend. Circ. Mat. Palermo (2) 1, 81--97 (1981; Zbl 0503.42018)]. When \(\mathcal P\) is a unit square with center at the origin, \(T_\varepsilon^\delta\) was studied by \textit{P. Oswald} [Moscow Univ. Math. Bull. 38, 65--73 (1983); translation from Vestn. Mosk. Univ., Ser. I 1983, No. 6, 56--63 (1983; Zbl 0531.42007) and Math. Nachr. 133, 173--187 (1987; Zbl 0641.42020)]. The authors' results generalize Oswald's case.
scientific article; zbMATH DE number 5149569
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