Riesz means associated with certain product type convex domain
Let \(f\in{\mathcal S}(\mathbb R^n)\) and denote by \(\widehat f\) the Fourier transform of \(f\). For \(n\geq 3\) and \(\xi=(\xi_0, \xi_1, \dots, \xi_{n-2}) \in\mathbb R^2\times\mathbb R^{n-2}\), define \(\rho(\xi):=\max\{|\xi_0|, |\xi_1|, \dots, |\xi_{n-2}|\}\). For \(\delta\in(0,\infty)\), define the convolution operator \(T_\varepsilon^\delta\) by \(\widehat{T_\varepsilon^\delta f}(x) :=(1-\frac{\rho(\xi)}{\varepsilon})_+^\delta\widehat f(\xi)\), \(T^\delta:=T_1^\delta\) and the maximal operator \(T_*^\delta\) by \[ T_*^\delta f(x) :=\sup_{\varepsilon>0}|T_\varepsilon^\delta f(x)|,\quad x=(x_0, x_1,\dots, x_{n-2})\in\mathbb R^2\times\mathbb R^{n-2},\;n\geq 3, \] where \(t_+^\delta:=t^\delta\) for \(t\in(0,\infty)\) and 0 otherwise. The authors show that there exists a positive constant \(C\) independent of \(f\) such that for all \(f\in H^p(\mathbb R^n)\), \(\|T_*^\delta f\|_{L^p(\mathbb R^n)} \leq C\|f\|_{H^p(\mathbb R^n)}\) if and only if \(p\in(\frac{2(n-1)}{2n-1}, 1)\) and \(\delta\in(n(\frac{1}{p}-1)+\frac{1}{2}, \infty)\), where \(H^p(\mathbb R^n)\) stands for the classical Hardy spaces. When \(p\in(\frac{2(n-1)}{2n-1}, 1)\) and \(\delta\in(n(\frac{1}{p}-1)+\frac{1}{2}, \infty)\) or \(p=\frac{2(n-1)}{2n-1}\) and \(\delta\in(n(\frac{1}{p}-1)+\frac{1}{2}, \infty)\), the authors show that \(|\{x\in\mathbb R^n:\;T_*^\delta (f)(x)>\alpha\}|\leq C\alpha^{-p}\|f\|_{H^p(\mathbb R^n)}^p\), where the positive constant \(C\) is independent of \(f\) or \(\alpha\). Finally, the authors show that the convolution operator \(T^\delta\) is bounded from \(L^p(\mathbb R^n)\) to \(L^p(\mathbb R^n)\) if and only if \(\delta>\max\{2|\frac{1}{p}-\frac{1}{2}|-\frac{1}{2}, 0\}\) and \(p\in[1,\infty)\). The authors also prove that if \(p=\frac {2(n-1)}{2n-1}\) and \(\delta = \frac {2n-1}{2(n-1)}\) or if \(p=1\) and \(\delta=\frac 12\), then \(T^\delta\) is unbounded from \(H^p(\mathbb R^n)\) to weak-\(L^p(\mathbb R^n)\).
scientific article; zbMATH DE number 5888547
scientific article