SOME OPERATORS ACTING ON WEIGHTED SEQUENCE BESOV SPACES AND APPLICATIONS

Abstract

In this article, we study the boundedness of matrix operators acting on weighted sequence Besov spaces $\dot{b}_{p,w}^{\alpha,q}$. First we obtain the necessary and sufficient condition for the boundedness of diagonal matrices acting on weighted sequence Besov space $\dot{b}_{p,w}^{\alpha,q}$, and investigate the duals of $\dot{b}_{p,w}^{\alpha,q}$, where the weight is non-negative and locally integrable. In particular, when $0<p< 1$, we find a type of new sequence sapces which characterize the dual space of $\dot{b}^{\alpha,q}_{p,w}$. We also use the duals of $\dot{b}_{p,w}^{\alpha,q}$ to characterize an algebra of matrix operators acting on weighted sequence Besov spaces $\dot{b}_{p,w}^{\alpha,q}$ and find the necessary and sufficient conditions to such a characterization. Note that we do not require that the given weight satisfies the doubling condition in this situation. Using these results, we give some applications to characterize the boundedness of Fourier-Haar multipliers and paraproduct operators. In this situation, we need to require that the weight $w$ is an $A_p$ weight.