arXiv subjects

Stephen Deterding

Publications and source records attributed to Stephen Deterding.

10 recordsLinked to original sources

arXiv2025

Geometric conditions for bounded point evaluations in several complex variables

Let $U$ be a bounded domain in $\mathbb C^d$ and let $L^p_a(U)$, $1 \leq p < \infty$, denote the space of functions that are analytic on $\overline{U}$ and bounded in the $L^p$ norm on $U$. A point $x \in \overline{U}$ is said to be a bounded point evaluation for $L^p_a(U)$ if the linear functional $f \to f(x)$ is bounded in $L^p_a(U)$. In this paper, we provide a purely geometric condition given in terms of the Sobolev $q$-capacity for a point to be a bounded point evaluation for $L^p_a(U)$. This extends results known only for the single variable case to several complex variables.

math.CV↗

arXiv2025

Analytic structure in spaces of Lipschitz functions

Let $U \subseteq \mathbb C$ be bounded and open. For $0 < \alpha < 1$, $A_\alpha(U)$ is the set of functions in the little Lipschitz class with exponent $\alpha$ that are analytic in a neighborhood of $U$. We consider three conditions, motivated by the properties of bounded point derivations, that show how the functions in $A_\alpha(U)$ can have additional analytic structure than would otherwise be expected. We prove an implication between conditions $(c)$ and $(b)$ and show that there is no implication between conditions $(a)$ and $(c)$.

math.CV↗

arXiv2024

Approximate Taylor theorem for analytic Lipschitz functions

Let $U$ be a bounded open subset of the complex plane and let $A_{\alpha}(U)$ denote the set of functions analytic on $U$ that also belong to the little Lipschitz class with Lipschitz exponent $\alpha$. It is shown that if $A_{\alpha}(U)$ admits a bounded point derivation at $x \in \partial U$, then there is an approximate Taylor Theorem for $A_{\alpha}(U)$ at $x$. This extends and generalizes known results concerning bounded point derivations.

math.CV↗

arXiv2023

Bounded point derivations on Campanato spaces

Let $X$ be a compact subset of the complex plane and $x \in X$. A necessary and sufficient condition is given in terms of Hausdorff contents for the existence of a bounded point derivation at $x$ on the space of vanishing Campanato functions that are analytic in a neighborhood of $X$. This generalizes many known conditions for the existence of bounded point derivations on other function spaces.

math.CV↗

arXiv2021

Boundary Smoothness conditions for functions in $R^p(X)$

Let $X$ be a compact subset of the complex plane and let $R^p(X)$, $2< p < \infty$, denote the closure of the rational functions with poles off $X$ in the $L^p$ norm. In this paper we consider three conditions that show how the functions in $R^p(X)$ can have a greater degree of smoothness at the boundary of $X$ than might otherwise be expected. We will show that two of the conditions are equivalent and imply the third but the third does not imply the other two.

math.CV↗

arXiv2020

H\"older conditions and $\tau$-spikes for analytic Lipschitz functions

Let $U$ be an open subset of $\mathbb{C}$ with boundary point $x_0$ and let $A_{\alpha}(U)$ be the space of functions analytic on $U$ that belong to lip$\alpha(U)$, the "little Lipschitz class". We consider the condition $S= \displaystyle \sum_{n=1}^{\infty}2^{(t+\lambda+1)n}M_*^{1+\alpha}(A_n \setminus U)< \infty,$ where $t$ is a non-negative integer, $0<\lambda<1$, $M_*^{1+\alpha}$ is the lower $1+\alpha$ dimensional Hausdorff content, and $A_n = \{z: 2^{-n-1}<|z-x_0|<2^{-n}\}$. This is similar to a necessary and sufficient condition for bounded point derivations on $A_{\alpha}(U)$ at $x_0$. We show that $S= \infty$ implies that $x_0$ is a $(t+\lambda)$-spike for $A_{\alpha}(U)$ and that if $S<\infty$ and $U$ satisfies a cone condition, then the $t$-th derivatives of functions in $A_{\alpha}(U)$ satisfy a H\"older condition at $x_0$ for a non-tangential approach.

math.FA↗

arXiv2019

Bounded point derivations and functions of bounded mean oscillation

Let $X$ be a subset of the complex plane and let $A_0(X)$ denote the space of VMO functions that are analytic on $X$. $A_0(X)$ is said to admit a bounded point derivation of order $t$ at a point $x_0 \in \partial X$ if there exists a constant $C$ such that $|f^{(t)}(x_0)|\leq C ||f||_{BMO}$ for all functions in $VMO(X)$ that are analytic on $X \cup \{x_0\}$. In this paper, we give necessary and sufficient conditions in terms of lower $1$-dimensional Hausdorff content for $A_0(X)$ to admit a bounded point derivation at $x_0$. These conditions are similar to conditions for the existence of bounded point derivations on other functions spaces.

math.CV↗

arXiv2018

Non-tangential limits for analytic Lipschitz functions

Let $U$ be a bounded open subset of the complex plane. Let $0<\alpha<1$ and let $A_{\alpha}(U)$ denote the space of functions that satisfy a Lipschitz condition with exponent $\alpha$ on the complex plane, are analytic on $U$ and are such that for each $\epsilon >0$, there exists $\delta >0$ such that for all $z$, $w \in U$, $|f(z)-f(w)| \leq \epsilon |z-w|^{\alpha}$ whenever $|z-w| < \delta$. We show that if a boundary point $x_0$ for $U$ admits a bounded point derivation for $A_{\alpha}(U)$ and $U$ has an interior cone at $x_0$ then one can evaluate the bounded point derivation by taking a limit of a difference quotient over a non-tangential ray to $x_0$. Notably our proofs are constructive in the sense that they make explicit use of the Cauchy integral formula.

math.CV↗

arXiv2017

A formula for a bounded point derivation on $R^p(X)$

Let $X$ be a compact subset of the complex plane. It is shown that if a point $x_0$ admits a bounded point derivation on $R^p(X)$, the closure of rational function with poles off $X$ in the $L^p(dA)$ norm, for $p >2$ and if $X$ contains an interior cone, then the bounded point derivation can be represented by the difference quotient if the limit is taken over a non-tangential ray to $x_0$. A similar result is proven for higher order bounded point derivations. These results extend a theorem of O'Farrell for $R(X)$, the closure of rational functions with poles off $X$ in the uniform norm.

math.CV↗

arXiv2017

Bounded point derivations on $R^p(X)$ and approximate derivatives

It is shown that if a point $x_0$ admits a bounded point derivation on $R^p(X)$, the closure of rational function with poles off $X$ in the $L^p(dA)$ norm, for $p >2$, then there is an approximate derivative at $x_0$. A similar result is proven for higher order bounded point derivations. This extends a result of Wang which was proven for $R(X)$, the uniform closure of rational functions with poles off $X$.

math.CV↗