We define a class of spaces $H_{\mu }^p$, 0 < p < ∞, of holomorphic functions on the tube, with a norm of Hardy type: ${\left|\right|F\left|\right|}_{H_{\mu }^p}^p=su{p}_{y\in \Omega }{\int }_{\Omega ̅}{\int }_{ℝⁿ}{\left|F(x+i(y+t))\right|}^{p}dxd\mu \left(t\right)$. We allow μ to be any quasi-invariant measure with respect to a group acting simply transitively on the cone. We show the existence of boundary limits for functions in $H_{\mu }^p$, and when p ≥ 1, characterize the boundary values as the functions in $L_{\mu }^p$ satisfying the tangential CR equations. A careful description of the measures μ when their supports lie on the boundary of the cone is also provided....

In this paper we define, by duality methods, a space of ultradistributions ${}_{\omega }^{\text{'}}\left({ℝ}^N\right)$. This space contains all tempered distributions and is closed under derivatives, complex translations and Fourier transform. Moreover, it contains some multipole series and all entire functions of order less than two. The method used to construct ${𝔾}_{\omega }^{\text{'}}\left({ℝ}^N\right)$ led us to a detailed study, presented at the beginning of the paper, of the duals of infinite dimensional locally convex spaces that are inductive limits of finite dimensional...

The notion of a $J^3$-triple is studied in connection with a geometrical approach to the generalized Hurwitz problem for quadratic or bilinear forms. Some properties are obtained, generalizing those derived earlier by the present authors for the Hurwitz maps S × V → V. In particular, the dependence of each scalar product involved on the symmetry or antisymmetry is discussed as well as the configurations depending on various choices of the metric tensors of scalar products of the basis elements. Then...

Using a construction similar to an iterated function system, but with functions changing at each step of iteration, we provide a natural example of a continuous one-parameter family of holomorphic functions of infinitely many variables. This family is parametrized by the compact space of positive integer sequences of prescribed growth and hence it can also be viewed as a parametric description of a trivial analytic multifunction.

We prove that for a parabolic subgroup $\Gamma$ of $\mathrm{Aut}{𝔹}^n$ the fixed points sets of all elements in $\Gamma \setminus \left\{{\mathrm{id}}_{{𝔹}^n}\right\}$ are the same. This result, together with a deep study of the structure of subgroups of $\mathrm{Aut}{𝔹}^n$ acting freely and properly discontinuously on ${𝔹}^n$, entails a generalization of the so called weak Hurwitz’s theorem: namely that, given a complex manifold $X$ covered by ${𝔹}^n$ and such that the group of deck transformations of the covering is “sufficiently generic”, then ${\mathrm{id}}_X$ is isolated in $\mathrm{Hol}(X,X)$.

Suppose $\varphi$ is a real analytic plurisubharmonic exhaustion function on a connected noncompact complex manifold $X$. The main result is that if the real analytic set of points at which $\varphi$ is not strongly $q$-convex is of dimension at most $2q+1$, then almost every sufficiently large sublevel of $\varphi$ is strongly $q$-convex as a complex manifold. For $X$ of dimension $2$, this is a special case of a theorem of Diederich and Ohsawa. A version for $\varphi$ real analytic with corners is also obtained.

The Leitmotiv of this work is to find suitable notions of dual varieties in a general sense. We develop the basic elements of a duality theory for varieties and complex spaces, by adopting a geometric and a categorical point of view. One main feature is to prove a biduality property for each notion which is achieved in most cases.

For smooth bounded pseudoconvex domains in ${ℂ}^2$, we provide geometric conditions on the boundary which imply compactness of the $\overline{\partial }$-Neumann operator. It is noteworthy that the proof of compactness does not proceed via verifying the known potential theoretic sufficient conditions.