Inequalities for Poisson integrals with slowly growing dimensional constants

Let $P_t$ be the Poisson kernel. We study the following $L^p$
inequality for the Poisson integral $Pf(x,t) =(P_t*f)(x)$
with respect to a Carleson measure $\mu$:
$$||Pf||_{L^p(\R_{+}^{n+1}, d \mu)} \le c_{p,n}
\kappa(\mu)^{{\frac1p}} ||f||_{L^p(\R^{n},dx)},$$

where $1 < p < \infty$ and $\kappa(\mu) $ is the Carleson norm of $\mu$.
It was shown by Verbitsky that for $p>2$ the constant
$c_{p,n}$ can be taken to be independent of the dimension $n$.
We show that $c_{2,n}=O((\log n )^{\frac12})$ and that
$c_{p,n} =O(n^{{\frac12}-{\frac1p}})$ for $1 < p < 2$ as
$n \to \infty$. We observe that standard proofs of this inequality
rely on doubling properties of cubes and lead to a value of $c_{p,n}$
that grows exponentially with $n$.