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A note on differentials of holomorphic functions

Richard Aron, Verónca Dimant, Manuel Maestre

Abstract

Recently, in arXiv:2304.07149, a bridge was made between the very active area of spaces of Lipschitz real functions on a metric space and holomorphic functions on an open subset of a Banach space. This was done by introducing and studying the space $\mathcal HL_0(B_X)$ of holomorphic Lipschitz functions defined on $B_X$, the open unit ball of the complex Banach space $X$ vanishing at 0. There it was proved that this space is isometrically isomorphic to a subspace of $\mathcal H^\infty(B_X, X^*)$, the space of bounded holomorphic mapping with values in the topological dual of $X$. In that paper it was shown that this subspace was a proper one, except in the one dimensional case. The goal of this note is to give an intrinsic characterization of the elements of that subspace. Moreover, in the case where $X$ additionally has a Schauder basis, it is shown that there is an explicit way to calculate whether and element of $\mathcal H^\infty(B_X, X^*)$ belongs or not to that subspace.

A note on differentials of holomorphic functions

Abstract

Recently, in arXiv:2304.07149, a bridge was made between the very active area of spaces of Lipschitz real functions on a metric space and holomorphic functions on an open subset of a Banach space. This was done by introducing and studying the space of holomorphic Lipschitz functions defined on , the open unit ball of the complex Banach space vanishing at 0. There it was proved that this space is isometrically isomorphic to a subspace of , the space of bounded holomorphic mapping with values in the topological dual of . In that paper it was shown that this subspace was a proper one, except in the one dimensional case. The goal of this note is to give an intrinsic characterization of the elements of that subspace. Moreover, in the case where additionally has a Schauder basis, it is shown that there is an explicit way to calculate whether and element of belongs or not to that subspace.
Paper Structure (4 sections, 10 theorems, 41 equations)

This paper contains 4 sections, 10 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Finite-dimensional case
  3. A general description
  4. Vector-valued case

Key Result

Lemma 2.1

Let $U\subset \mathbb{C}^n$ be an open star-like set with respect to a point $a\in U$ and let $f:U\to\mathbb{C}$ be a holomorphic function. Then, the mapping $F=(F_1,\dots , F_n):U\to\mathbb{C}^n$ defined by is differentiable with partial derivatives given by for all $k\in\{1,\dots, n\}\setminus \{j\}$ and

Theorems & Definitions (20)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • Example 2.4
  • Remark 3.1
  • Proposition 3.2
  • proof
  • ...and 10 more