Optimal Error Estimates for Analytic Continuation in the Upper Half‐Plane

Abstract

AbstractAnalytic functions in the Hardy class H2 over the upper half‐plane ℍ+ are uniquely determined by their values on any curve Γ lying in the interior or on the boundary of ℍ+. The goal of this paper is to provide a sharp quantitative version of this statement. We answer the following question: Given f of a unit H2‐norm that is small on Γ (say, its L2‐norm is of order ϵ), how large can f be at a point z away from the curve? When Γ ⊂ ∂ℍ+, we give a sharp upper bound on ∣f(z)∣ of the form ϵγ, with an explicit exponent γ = γ(z) ∈ (0, 1) and explicit maximizer function attaining the upper bound. When Γ ⊂ ℍ+ we give an implicit sharp upper bound in terms of a solution of an integral equation on Γ. We conjecture and give evidence that this bound also behaves like ϵγ for some γ = γ(z) ∈ (0, 1). These results can also be transplanted to other domains conformally equivalent to the upper half‐plane. © 2020 Wiley Periodicals, Inc.