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

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.

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