  We consider the nonparametric regression with a random design model, and we
are interested in the adaptive estimation of the regression at a point $x\_0$
where the design is degenerate. When the design density is $\beta$-regularly
varying at $x\_0$ and $f$ has a smoothness $s$ in the H\"{o}lder sense, we know
from Ga\"{i}ffas (2004)\nocite{gaiffas04a} that the minimax rate is equal to
$n^{-s/(1+2s+\beta)} \ell(1/n)$ where $\ell$ is slowly varying. In this paper
we provide an estimator which is adaptive both on the design and the regression
function smoothness and we show that it converges with the rate $(\log
n/n)^{s/(1+2s+\beta)} \ell(\log n/n)$. The procedure consists of a local
polynomial estimator with a Lepski type data-driven bandwidth selector similar
to the one in Goldenshluger and Nemirovski
(1997)\nocite{goldenshluger\_nemirovski97} or Spokoiny (1998)\nocite{spok98}.
Moreover, we prove that the payment of a $\log$ in this adaptive rate compared
to the minimax rate is unavoidable.
