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 ${\mathcal{H}\mathcal{T}}_{q,\alpha }^p$ into the tent spaces $T_s^t(\mu )$ is bounded for all $0<p,q,s,t<\mathrm{\infty }$ and $\alpha >-n-1$. As an application, we determine the indices $0<p,q,s,t<\mathrm{\infty }$ and $\alpha ,\beta >-n-1$ such that the inclusion ${\mathcal{H}\mathcal{T}}_{q,\alpha }^p\subset {\mathcal{H}\mathcal{T}}_{s,\beta }^t$ holds, which allows us to completely characterize the nonlinear superposition operators between Hardy type tent spaces.