  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.
