Weak type estimates for maximal operators over convex hypersurfaces
Let \(\Omega\) be a convex domain in \(\mathbb R^2\) with smooth boundary which contains the origin in its interior, and \(\partial \Omega\) is a convex hypersurface of finite type. Let \(\rho\) be the Minkowski functional associated with \(\Omega\), so \(\rho(t\xi)=t\rho(\xi)\) and \(\rho(\xi)=1\) for \(\xi\in\partial \Omega\). The author considers a rough distance function \[ m(\xi',\xi_3)=\max\{\rho(\xi'),| \xi_3| \},\quad (\xi',\xi_3)\in\mathbb R^2 \times\mathbb R, \] and defines convolution operators \[ S_\varepsilon^\delta f(x,t)=\mathcal F^{-1}\bigl[(1-m/\varepsilon)_+^\delta \hat f \bigr](x,t), \quad (x,t)\in \mathbb R^2\times\mathbb R, \] and the maximal operator \[ S_*^\delta f(x,t)=\sup_{\varepsilon>0}S_\varepsilon^\delta f(x,t). \] He gives the following: 1. The maximal operator \(S_*^\delta\) is bounded from \(H^p(\mathbb R^3)\) to \(L^p(\mathbb R^3)\), if and only if \(4/5<p\leq 1\) and \(\delta>\delta(p):=3(1/p-1)+1/2\). 2. If \(4/5<p\leq 1\) and \(\delta=\delta(p)\), or \(p=4/5\) and \(\delta>\delta(p)\), then \(S_*^\delta\) maps \(H^p(\mathbb R^3)\) boundedly to weak-\(L^p(\mathbb R^3)\). These results are known in the case \(\rho(\xi)=| \xi| \) [\textit{S. Hong} et al., Math. Z. 253, No. 1, 1--24 (2006; Zbl 1105.42007)]). Several optimal results are discussed, and the \(n\)-dimensional case (\(n\geq 3\)) is also discussed.
scientific article; zbMATH DE number 6121087
scientific article