A binomial residue is a rational function defined by a hypergeometric integral whose kernel is singular along binomial divisors. Binomial residues provide an integral representation for rational solutions of $A$-hypergeometric systems of Lawrence type. The space of binomial residues of a given degree, modulo those which are polynomial in some variable, has dimension equal to the Euler characteristic of the matroid associated with $A$.

We initiate the study of Bloch type spaces on the unit ball of a Hilbert space. As applications, the Hardy-Littlewood theorem in infinite-dimensional Hilbert spaces and characterizations of some holomorphic function spaces related to the Bloch type space are presented.

Let φ be a holomorphic mapping between complex unit balls. We characterize those regular φ for which the composition operators C φ: f ↦ f ○ φ map the Bloch space into the Hardy space.

Let $S^{\text{'}}$ be the class of tempered distributions. For $f\in S^{\text{'}}$ we denote by $J^{-\alpha }f$ the Bessel potential of $f$ of order $\alpha$. We prove that if $J^{-\alpha }f\in \mathrm{B}MO$, then for any $\lambda \in (0,1)$, $J^{-\alpha }{\left(f\right)}_{\lambda }\in \mathrm{B}MO$, where ${\left(f\right)}_{\lambda }={\lambda }^{-n}f\left(\phi ({\lambda }^{-1}·)\right)$, $\phi \in S$. Also, we give necessary and sufficient conditions in order that the Bessel potential of a tempered distribution of order $\alpha >0$ belongs to the $\mathrm{V}MO$ space.

We survey variations of the Bergman kernel and their asymptotic behaviors at degeneration. For a Legendre family of elliptic curves, the curvature form of the relative Bergman kernel metric is equal to the Poincaré metric on ℂ 0,1. The cases of other elliptic curves are either the same or trivial. Two proofs depending on elliptic functions’ special properties and Abelian differentials’ Taylor expansions are discussed, respectively. For a holomorphic family of hyperelliptic nodal or cuspidal curves...

We study the boundary behaviour of holomorphic functions in the Hardy-Sobolev spaces ${\mathscr{H}}^{p,k}\left(\right)$, where is a smooth, bounded convex domain of finite type in ℂⁿ, by describing the approach regions for such functions. In particular, we extend a phenomenon first discovered by Nagel-Rudin and Shapiro in the case of the unit disk, and later extended by Sueiro to the case of strongly pseudoconvex domains.

We solve the Dirichlet problem for line integrals of holomorphic functions in the unit ball: For a function $u$ which is lower semi-continuous on $\partial {𝔹}^n$ we give necessary and sufficient conditions in order that there exists a holomorphic function $f\in 𝕆\left({𝔹}^n\right)$ such that $$u\left(z\right)={\int }_{\left|\lambda \right|<1}{\left|f\left(\lambda z\right)\right|}^2\mathrm{d}{𝔏}^2\left(\lambda \right).$$

In strictly pseudoconvex domains with smooth boundary, we prove a commutator relationship between admissible integral operators, as introduced by Lieb and Range, and smooth vector fields which are tangential at boundary points. This makes it possible to gain estimates for admissible operators in function spaces which involve tangential derivatives. Examples are given under with circumstances these can be transformed into genuine Sobolev- and Ck-estimates.