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

Analytic functions in the Hardy class H² over the upper half‐plane ?₊ are uniquely determined by their values on any curve Γ lying in the interior or on the boundary of ?₊ . 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 ? ), how large can f be at a point z away from the curve? When Γ ? ??₊ , we give a sharp upper bound on ∣f(z)∣ of the form ?^γ , with an explicit exponent γ = γ(z) ∈ (0, 1) and explicit maximizer function attaining the upper bound. When Γ ? ?₊ 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 ?^γ 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.