# Optimal error estimates for analytic continuation in the upper half-plane

## TL;DR

This paper provides sharp quantitative bounds on how small values of Hardy class functions on a curve influence their magnitude at interior points, with explicit exponents and maximizers, advancing understanding of analytic continuation error estimates.

## Contribution

It offers the first sharp, explicit error bounds for analytic continuation in the upper half-plane, including maximizer functions and integral equation characterizations.

## Key findings

- Sharp upper bounds of the form ε^γ for functions on boundary curves
- Implicit bounds involving integral equations for interior curves
- Identification of maximizer functions attaining the bounds

## Abstract

Analytic functions in the Hardy class $H^2$ over the upper half-plane $\mathbb{H}_+$ are uniquely determined by their values on any curve $\Gamma$ lying in the interior or on the boundary of $\mathbb{H}_+$. The goal of this paper is to provide a quantitative version of this statement. Given that $f$ from a unit ball in $H^2$ is small on $\Gamma$ (say, its $L^2$ norm is of order $\epsilon$), how does this affect the magnitude of $f$ at a point $z$ away from the curve? When $\Gamma \subset \partial \mathbb{H}_+$, we give a sharp upper bound on $|f(z)|$ of the form $\epsilon^\gamma$, with an explicit exponent $\gamma=\gamma(z) \in (0,1)$ and describe the maximizer function attaining the upper bound. When $\Gamma \subset \mathbb{H}_+$ we give an implicit sharp upper bound in terms of a solution of an integral equation on $\Gamma$. We conjecture and give evidence that this bound also behaves like $\epsilon^\gamma$ for some $\gamma=\gamma(z) \in (0,1)$. These results can also be transplanted to other domains conformally equivalent to the upper half-plane.