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Analytic functions in the Hardy class   H  2   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   H  2  ‐norm that is small on  Γ  (say, its   L  2  ‐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   ϵ   γ   , 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   ϵ   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.

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