Let $K$ be a compact set in an open set $D$ on a Stein manifold $\Omega$ of dimension $n$. We denote by $H^{\infty }\left(D\right)$ the Banach space of all bounded and analytic in $D$ functions endowed with the uniform norm and by $A_K^D$ a compact subset of the space $C\left(K\right)$ consisted of all restrictions of functions from the unit ball ${𝔹}_{H^{\infty }\left(D\right)}$. In 1950ies Kolmogorov posed a problem: does$${\mathscr{H}}_{\epsilon }\left(A_K^D\right)\sim \tau {\left(ln\frac{1}{\epsilon }\right)}^{n+1},\epsilon \to 0,$$where ${\mathscr{H}}_{\epsilon }\left(A_K^D\right)$ is the $\epsilon$-entropy of the compact $A_K^D$. We give here a survey of results concerned with this problem and a related problem on the strict asymptotics of Kolmogorov diameters...

We study the holomorphic Hardy-Orlicz spaces ${}^{\Phi }\left(\Omega \right)$, where Ω is the unit ball or, more generally, a convex domain of finite type or a strictly pseudoconvex domain in ℂⁿ. The function Φ is in particular such that $¹\left(\Omega \right){\subset }^{\Phi }\left(\Omega \right){\subset }^p\left(\Omega \right)$ for some p > 0. We develop maximal characterizations, atomic and molecular decompositions. We then prove weak factorization theorems involving the space BMOA(Ω). As a consequence, we characterize those Hankel operators which are bounded from ${}^{\Phi }\left(\Omega \right)$ into ¹(Ω).

This work is an introduction to anisotropic spaces of holomorphic functions, which have $\omega$-weight and are generalizations of Bloch spaces on a unit ball. We describe the holomorphic Bloch space in terms of the corresponding $L_{\omega }^{\infty }$ space. We establish a description of ${\left(A^p\left(\omega \right)\right)}^{*}$ via the Bloch classes for all $0<p\le 1$.

We construct a function f holomorphic in a balanced domain D in ${ℂ}^N$ such that for every positive-dimensional subspace Π of ${ℂ}^N$, and for every p with 1 ≤ p < ∞, ${f|}_{\Pi \cap D}$ is not $L^p$-integrable on Π ∩ D.

Let D be a bounded strictly pseudoconvex domain of ${ℂ}^n$ with smooth boundary. We consider the weighted mixed-norm spaces $A_{\delta ,k}^{p,q}\left(D\right)$ of holomorphic functions with norm $\parallel f{\parallel }_{p,q,\delta ,k}=({\sum }_{\left|\alpha \right|\le k}{ʃ}_0^{r_0}({ʃ}_{\partial D_r}|D^{\alpha }{f|}^{p}d{\sigma }_r{{)}^{q/p}{r}^{\delta q/p-1}dr)}^{1/q}$. We prove that these spaces can be obtained by real interpolation between Bergman-Sobolev spaces $A_{\delta ,k}^p\left(D\right)$ and we give results about real and complex interpolation between them. We apply these results to prove that $A_{\delta ,k}^{p,q}\left(D\right)$ is the intersection of a Besov space $B_s^{p,q}\left(D\right)$ with the space of holomorphic functions on D. Further, we obtain several properties of the mixed-norm...

Besov spaces of holomorphic functions in tubes over cones have been recently defined by Békollé et al. In this paper we show that Besov p-seminorms are invariant under conformal transformations of the domain when n/r is an integer, at least in the range 2-r/n < p ≤ ∞.

We show that T is a surjective multiplicative (but not necessarily linear) isometry from the Smirnov class on the open unit disk, the ball, or the polydisk onto itself, if and only if there exists a holomorphic automorphism Φ such that T(f)=f ○ Φ for every class element f or T(f) = $\overline{f^{\circ }\overline{\varphi }}$ for every class element f, where the automorphism Φ is a unitary transformation in the case of the ball and Φ(z 1, ..., z n) = $({\lambda }_1{z}_{i_1},...,{\lambda }_n{z}_{i_n})$ for |λ j| = 1, 1 ≤ j ≤ n, and (i 1; ..., i n)is some permutation of the integers from...

For a given direction $𝐛\in {ℂ}^n\setminus \left\{\mathbf{0}\right\}$ we study non-homogeneous directional linear higher-order equations whose all coefficients belong to a class of joint continuous functions which are holomorphic on intersection of all directional slices with a unit ball. Conditions are established providing boundedness of $L$-index in the direction with a positive continuous function $L$ satisfying some behavior conditions in the unit ball. The provided conditions concern every solution belonging to the same class of functions...