### On some characterizations of Carleson type measure in the unit ball.

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For any holomorphic function $f$ on the unit polydisk ${\mathbb{D}}^{n}$ we consider its restriction to the diagonal, i.e., the function in the unit disc $\mathbb{D}\subset \u2102$ defined by $\mathrm{Diag}f\left(z\right)=f(z,...,z)$, and prove that the diagonal map $\mathrm{Diag}$ maps the space ${Q}_{p,q,s}\left({\mathbb{D}}^{n}\right)$ of the polydisk onto the space ${\widehat{Q}}_{p,s,n}^{q}\left(\mathbb{D}\right)$ of the unit disk.

We provide new assertions on factorization of tent spaces.

We present a description of the diagonal of several spaces in the polydisk. We also generalize some previously known contentions and obtain some new assertions on the diagonal map using maximal functions and vector valued embedding theorems, and integral representations based on finite Blaschke products. All our results were previously known in the unit disk.

We present new sharp embedding theorems for mixed-norm analytic spaces in pseudoconvex domains with smooth boundary. New related sharp results in minimal bounded homogeneous domains in higher dimension are also provided. Last domains we consider are domains which are direct generalizations of the well-studied so-called bounded symmetric domains in ${\u2102}^{n}.$ Our results were known before only in the very particular case of domains of such type in the unit ball. As in the unit ball case, all our proofs are...

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