Embedding Hardy spaces $H^p$ into tent spaces and generalized integration operators

Let $0 \lt p,s \lt \infty $, $n$ be a nonnegative integer and $\mu $ be a positive Borel measure on $\mathbb {D}$. Let $\mathcal {T}_{s}^{p,n}{(\mu )}$ be the space of all analytic functions $f $ such that $$\sup _{I\subseteq \partial \mathbb D}\frac {1}{|I|^s}\int _{S(I)}|f^{(n)}(z)(1-|z|^2)^n|^p\,d\mu (z) \lt \infty .$$ In this paper, the boundedness and compactness of embedding from Hardy spaces $H^p$ into $\mathcal {T}_{s}^{p,n}{(\mu )}$ are studied. As an application, the boundedness, compactness and essential norm of the generalized integral operators $T_{g}^{n,k}$ and $S_{g}^{n,0}$ acting from $H^p$ to general function spaces are also investigated.