Equivalent Characterizations and Pointwise Multipliers of Normal Weight Dirichlet Space on the Unit Ball

Abstract

Let $$\mu $$ be a normal function on [0, 1) and $$p>0$$ . In this paper, the authors give several equivalent characterizations of the functions in the normal weight Dirichlet type space $$D_{\mu }^{p}(B)$$ on the unit ball B in $$\mathbf{C}^{n}$$ . At the same time, the authors describe the boundedness of a class of integral operator T from the normal weight Lebesgue space $$L_{\mu }^{p}(B)$$ to the abstract measure Lebesgue space $$L_{\mu ,\psi }^{p}(B)$$ . As an application of the previous results, the authors discuss the pointwise multipliers on $$D_{\mu }^{p}(B)$$ or from $$D_{\mu }^{p}(B)$$ to $${\mathcal {B}}_{\nu _{p}}(B)$$ .