Tent Carleson measures and superposition operators on Hardy type tent spaces

In this paper, we completely characterize the positive Borel measures $\mu$ on the unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ such that the Carleson embedding from holomorphic Hardy type tent spaces $\mathcal{HT}^p_{q,\alpha}$ into the tent spaces $T^t_s(\mu)$ is bounded for all $0<p,q,s,t<\infty$ and $\alpha>-n-1$. As an application, we determine the indices $0<p,q,s,t<\infty$ and $\alpha,\beta>-n-1$ such that the inclusion $\mathcal{HT}^p_{q,\alpha}\subset\mathcal{HT}^t_{s,\beta}$ holds, which allows us to completely characterize the nonlinear superposition operators between Hardy type tent spaces.