The boundedness of commutators on locally compact Vilenkin groups (Q558510)

Theorems on locally compact Vilenkin groups \(G\) are proved. Theorem 1. Let \(1< p<\infty\), \(0<\beta< 1\), \(b\in \text{Lip}_\beta(G)\). If \(T\) is a singular integral operator, then the commutator \([b, T]f(x)= b(x)Tf(x)- T(bf)(x)\) is bounded from \(L^p(G)\) to \(\dot F^{\beta,\infty}_p(G)\). Theorem 2. Let \(1< q<\infty\), \(0<\beta< 1\), \(b\in\text{Lip}_\beta(G)\), \(0<\alpha<1-{1\over q}\). If \(T\) is a singular integral operator, then the commutator \([b,T]f(x)= b(x)Tf(x)- T(bf)(x)\) is bounded from \(\dot K^{\alpha,p}_p(G)\) to \(\dot K^{\alpha,p}_q \dot F^\infty_\beta(G)\). Theorem 3. Let \(1\leq r< q<\infty\). If a sublinear operator \(T\) satisfies \[ |Tf(x)|\leq C\Biggl(\int_K {|f(y)|^r\over|x- y|} dy\Biggr)^{{1\over r}},\quad x\not\in K, \] where \(f\in L^q(G)\) has compact support \(K= \text{supp\,}f\), and \(T: L^q(G)\to L^q(G)\), then \(T\) is bounded on \(\dot K^{\alpha,p}_p(G)\) provided \(0<\alpha<{1\over r}- {1\over q}\), \(0< p\leq\infty\). The proofs are technique.