Calderón-Zygmund operators on weighted weak Hardy spaces in locally compact Vilenkin groups (Q2732535)

The authors take a locally compact Vilenkin group \(G\) as basis space to study Calderón-Zygmund operators on weighted weak Hardy spaces. For \(1\leq p<\infty\), the Muckenhoupt class \(A_p(G)\) on \(G\) is defined and the space \(L^p_\omega (G)\) of all measurable functions with weight \(\omega\) on \(G\). For a locally integrable function and \(0<p <\infty\) the weighted weak Lebesgue spaces \(WL^p_\omega(G)\) as well as \(f^*\in S'(G)\) NEWLINE\[NEWLINEf^*(x)= \sup_{n\in \mathbb{Z}} \bigl|f^*\Delta_n (x)\bigr|NEWLINE\]NEWLINE with \(\Delta_n(x)=|G_n |^{-1} \chi_{G_n} (x)\), \(\chi_{G_n}(x)\) the characteristic function of \(G_n\), and the Haar measure \(|G_n|\) of \(G\) are considered. For \(p\in(0,1]\), the weighted weak Hardy spaces are defined by NEWLINE\[NEWLINEWH^p_\omega (G)=\bigl\{ f\in S'(G): \|f^*\|_{WL^p_\omega (G)} <\infty \bigr\}NEWLINE\]NEWLINE and \(\|f\|_{W H^p_\omega (G)}=\|f^*\|_{WL^p_\omega(G)}\).NEWLINENEWLINENEWLINESeven theorems are proved for the atom decomposition of \(WH^p_\omega(G)\), the boundedness properties of \(\delta\)-type Calderón-Zygmund operators and the boundedness properties of generalized Calderón-Zygmund operators.