Carleson measures, tent embeddings, and Volterra-type integral operators on the unit ball

In this paper, we establish a sharp comparison between Carleson-cube and Bergman-metric-ball conditions on the open unit ball $\text{B}$ and combine it with a Berezin-type characterization to prove embedding theorems for Besov spaces and Bergman spaces on $\text{B}$ into logarithmic tent spaces in the Bergman metric. As applications, we characterize the boundedness, compactness, and essential norms of the Volterra-type integral operators $T_g$ and $I_g$ acting from the Besov space $B_t(\text{B})$ to the general function space $F(p,q,s)$.