Herz-type Hardy spaces on Vilenkin groups and their applications
Let \(G\) be a locally compact abelian group containing a strictly decreasing sequence of compact open subgroups \(\{G_n\}_{n=-\infty}^{\infty}\) such that \(\bigcup_{n=-\infty}^{\infty}G_n=G\), \(\bigcap_{n=-\infty}^{\infty}G_n=\{0\}\) and \(\sup\{\text{order}(G_n/G_{n+1}): n\in \mathbb Z \}<\infty\). Such a space is called a locally compact Vilenkin group. Let \(dx\) be the Haar measure such that the Haar measure \(|G_0|\) of \(G_0\) is \(1\). \(G_n\) plays the role of the ball of radius \(n\) and of center at the origin in the Euclidean case. For nonnegative locally integrable functions \(\omega_1\), \(\omega_2\) on \(G\), \(0\leq \alpha <\infty\), and \(0<p, q<\infty\), the authors define the homogeneous Herz space \(\dot{K}_{q}^{\alpha, p}(\omega_1,\omega_2;G)\) by the set \(\{f\); \(f\) is a measurable function on \(G\) and \(\|f\|_{\dot{K}_{q}^{\alpha, p}(\omega_1,\omega_2;G)}<\infty \}\), where \(\|f\|_{\dot{K}_{q}^{\alpha, p}(\omega_1,\omega_2;G)}= \{\sum_{l=-\infty}^{\infty}[\omega_1(G_l)]^{\alpha p} \|f\chi_{G_l\setminus G_{l+1}}\|_{L_{\omega_2}^q(G)}^p\}^{1/p}\). The Herz-type Hardy spaces and local Hardy spaces are defined analogously to the Euclidean case. The authors also define central blocks and atoms analogously. Under appropriate assumptions on \(\omega_1\) and/or \(\omega_2\), they present central block or atom decompositions of these function spaces. Using these characterizations, they investigate the boundedness on the Herz-type Hardy spaces of the fractional integral operators and the boundedness on the local Herz-type Hardy spaces of pseudo-differential operators of order zero.
scientific article; zbMATH DE number 1531255
scientific article