  The aim of this paper is to recover the regression function with sup norm
loss. We construct an asymptotically sharp estimator which converges with the
spatially dependent rate r\_{n, \mu}(x) = P \big(\log n / (n \mu(x)) \big)^{s /
(2s + 1)}, where $\mu$ is the design density, $s$ the regression smoothness,
$n$ the sample size and $P$ is a constant expressed in terms of a solution to a
problem of optimal recovery as in Donoho (1994). We prove this result under the
assumption that $\mu$ is positive and continuous. This estimator combines
kernel and local polynomial methods, where the kernel is given by optimal
recovery, which allows to prove the result up to the constants for any $s > 0$.
Moreover, the estimator does not depend on $\mu$. We prove that $r\_{n,
\mu}(x)$ is optimal in a sense which is stronger than the classical minimax
lower bound. Then, an inhomogeneous confidence band is proposed. This band has
a non constant length which depends on the local amount of data.
