  Let $K\subset \mathbb R$ be a regular compact set and let
$g(z)=g_{\overline{\mathbb C}\setminus K}(z,\infty)$ be the Green function for
$\overline{\mathbb C}\setminus K$ with pole at infinity. For $\delta>0$, define
$$ G(\delta):=\max\{ g(z): z\in \mathbb C, \,\operatorname{dist}(z,K)\le
2\delta\}. $$ Let $\{ x_n\}_{n=0}^\infty$ be a Leja sequence of points of $K$.
Then the uniform norm $\|T_n\|=\Lambda_n, n=1,2,\ldots$ of the associated
interpolation operator $T_n$, i.e., the $n$-th Lebesgue constant, is bounded
from above by $$ \min_{\delta>0}2n\left[\frac{\operatorname{diam}(
K)}{\delta}e^{nG(\delta)}\right]^{9/8}. $$ In particular, when $K$ is a
uniformly perfect subset of $\mathbb R$, the Lebesgue constants grow at most
polynomially in $n$.
  To the best of our knowledge, the result is new even when $K$ is a finite
union of intervals.
