Littlewood-Paley and pseudo-differential operators on Herz-type spaces over Vilenkin groups
The author studies Littlewood-Paley and pseudo-differential operators on Herz-type spaces over Vilenkin groups. Suppose that \(G\) is a Vilenkin group with character group \(\Gamma\), the Littlewood-Paley operator on \(G\) is defined by \[ Sf(x)= \Biggl\{\sum_{n\in\mathbb{Z}} |(f^\wedge\chi_{\Gamma_{n+1}\setminus\Gamma_n})^\vee(x)|^2 \Biggr\}^{{1\over 2}},\quad x\in G, \] and the pseudo-differential operator on \(G\) is defined by \[ T_\sigma f(x)= \int_\Gamma\sigma(\gamma) f^\wedge(\gamma) \gamma(x)\, d\gamma \] with \(\sigma\in S^m_p\). Moreover, the Herz-type space on \(G\) is \(\dot K^{\alpha,p}_q(\omega_1,\omega_2; G)\) with norm \[ \| f\|_{\dot K^{\alpha,p}_q(\omega_1,\omega_2)}= \Biggl\{\sum^{+ \infty}_{l=-\infty} |\omega_1(G_1)|^{\alpha p}\| f\chi_{G_l\setminus G_{l+1}}\|^p_{L^q_{\omega_2(G)}}\Biggr\}^{{1\over p}}, \] and the Herz-type Hardy space on \(G\) is \[ H\dot K^{\alpha,p}_q(\omega_1, \omega_2; G)= \{f\in S'(G): f^*\in \dot K^{\alpha, p}_q(\omega_1, \omega_2; G)\} \] with \(f^*(x)= \sup_{n\in\mathbb{Z}}|f^*\Delta_n(x)|\), \(\| f\|_{H\dot K^{\alpha,p}_q(\omega_1, \omega_2; G)}= \| f^*\|_{\dot K^{\alpha,p}_q(\omega_1, \omega_2; G)}\). The main results are: 1. The operator \(S\) is bounded from \(H\dot K^{\alpha,p}_q(\omega_1, \omega_2; G)\) to \(\dot K^{\alpha, p}_q(\omega_1, \omega_2; G)\) for \(1< q< \infty\), \(0< p<\infty\), \(1-{1\over q}\leq \alpha< \infty\) where \(\omega_1,\omega_2\in A_1\) are certain nonnegative weights; 2. the operator \(T_\sigma\) is bounded on \(H\dot K^{\alpha,p}_q(\omega_1,\omega_2; G)\) for \(1< q<\infty\), \(0< p< 1\) and \(\alpha> \max(1-{1\over q},{1\over p}- 1)\) with the symbol \(\sigma\in S^m_p\), \(m\leq \alpha\), \(\rho\geq 1\).
scientific article; zbMATH DE number 2174190