A Pointwise Bound for a Holomorphic Function which is Square-Integrable with Respect to an Exponential Density Function

Holomorphic jets in symplectic manifolds

We define pointwise partial differential relations for holomorphic discs. Given a relative homotopy class, a relation, and a generic almost complex structure we provide the moduli space of discs which have an injective point with the structure of a smooth manifold. Applications to the local behaviour are given and an adjunction inequality for singularities is derived. Moreover, we show that for a coordinate class of a monotone Lagrangian split torus generically the number of non-immersed holomorphic discs is even.

Norms of embeddings between quadratically weighted spaces of holomorphic functions

We consider spaces of holomorphic functions which are square-integrable against a Gaussian weight, which appear in the theory of metaplectic FBI--Bargmann transforms. We identify the operator norm of embeddings between two such spaces, by relating these embeddings to Fourier integral operators with complex phase.

Local and global holomorphic extensions of time-varying real analytic vector fields

In this paper, we consider time-varying real analytic vector fields as curves on the space of real analytic vector fields. Using a suitable topology on the space of real analytic vector fields, we study and characterize different properties of time-varying real analytic vector fields. We study holomorphic extensions of time-varying real analytic vector fields and show that under suitable integrability conditions, a time-varying real analytic vector field on a manifold can be extended to a time-varying holomorphic vector field on a neighbourhood of that manifold. Moreover, we develop an operato…

Holomorphic functions of finite order generated by Dirichlet series

Given a frequency $λ= (λ_n)$ and $\ell \ge 0$, we introduce the scale of Banach spaces $H_{\infty,\ell}^λ[Re > 0]$ of holomorphic functions $f$ on the open right half-plane $[Re > 0]$, which satisfy $(A)$ the growth condition $|f(s)| = O((1 + |s|)^\ell)$, and $(B)$ have a Riesz germ, i.e. on some open subset and for some $m \ge 0$ the function $f$ coincides with the pointwise limit (as $x \to \infty$) of the so-called $(λ,m)$-Riesz means $\sum_{λ_n < x} a_n e^{-λ_n s}\big( 1-\frac{λ_n}{x}\big)^m ,\,x >0$ of some $λ$-Dirichlet series $\sum a_n e^{-λ_n s}$. Reformulated in our terminology, an im…

Henstock multivalued integrability in Banach lattices with respect to pointwise non atomic measures

Henstock-type integrals are considered, for multifunctions taking values in the family of weakly compact and convex subsets of a Banach lattice $X$. The main tool to handle the multivalued case is a Rådström-type embedding theorem established by C. C. A. Labuschagne, A. L. Pinchuck, C. J. van Alten in 2007. In this way the norm and order integrals reduce to that of a single-valued function taking values in an $M$-space, and new proofs are deduced for some decomposition results recently stated in two recent papers by Di Piazza and Musial based on the existence of integrable selections.

Constrained optimization in classes of analytic functions with prescribed pointwise values

We consider an overdetermined problem for Laplace equation on a disk with partial boundary data where additional pointwise data inside the disk have to be taken into account. After reformulation, this ill-posed problem reduces to a bounded extremal problem of best norm-constrained approximation of partial L2 boundary data by traces of holomorphic functions which satisfy given pointwise interpolation conditions. The problem of best norm-constrained approximation of a given L2 function on a subset of the circle by the trace of a H2 function has been considered in [Baratchart \& Leblond, 1998]. I…

Extension of Lipschitz-type operators on Banach function spaces

We study extension theorems for Lipschitz-type operators acting on metric spaces and with values on spaces of integrable functions. Pointwise domination is not a natural feature of such spaces, and so almost everywhere inequalities and other measure-theoretic notions are introduced.%more appropriate. Thus, we adapt the classical definition of Lipschitz map to the context of spaces of integrable functions by introducing such elements. We analyze Lipschitz type inequalities in two fundamental cases. The first concerns a.e. pointwise inequalities, while the second considers dominations involving …

特殊 null 序列的空性

Spaceability of special families of null sequences of holomorphic functions

本文研究了特殊 null 序列的空性，证明了两个闭合无限维子空间的存在，分别涉及点集收敛与紧致收敛的性质。

向量值全纯函数与抽象 Fubini 型定理

Vector-Valued Holomorphic Functions and Abstract Fubini-Type Theorems

针对取值于 L_p(Ω) 的向量值全纯函数，本文证明了线性泛函与对偶元作用可交换的抽象 Fubini 型恒等式。

The inversion formula and holomorphic extension of the minimal representation of the conformal group

The minimal representation $π$ of the indefinite orthogonal group $O(m+1,2)$ is realized on the Hilbert space of square integrable functions on $\mathbb R^m$ with respect to the measure $|x|^{-1} dx_1... dx_m$. This article gives an explicit integral formula for the holomorphic extension of $π$ to a holomorphic semigroup of $O(m+3, \mathbb C)$ by means of the Bessel function. Taking its `boundary value', we also find the integral kernel of the `inversion operator' corresponding to the inversion element on the Minkowski space $\mathbb R^{m,1}$.

Representation formulas and pointwise properties for Barron functions

We study the natural function space for infinitely wide two-layer neural networks with ReLU activation (Barron space) and establish different representation formulae. In two cases, we describe the space explicitly up to isomorphism. Using a convenient representation, we study the pointwise properties of two-layer networks and show that functions whose singular set is fractal or curved (for example distance functions from smooth submanifolds) cannot be represented by infinitely wide two-layer networks with finite path-norm. We use this structure theorem to show that the only $C^1$-diffeomorphis…

Describing limits of integrable functions as grid functions of nonstandard analysis

In functional analysis, there are different notions of limit for a bounded sequence of $L^1$ functions. Besides the pointwise limit, that does not always exist, the behaviour of a bounded sequence of $L^1$ functions can be described in terms of its weak-$\star$ limit or by introducing a measure-valued notion of limit in the sense of Young measures. Working in Robinson's framework of analysis with infinitesimals, we show that for every bounded sequence $\{z_n\}_{n \in \mathbb{N}}$ of $L^1$ functions there exists a function of a hyperfinite domain (i.e.\ a grid function) that represents both the…