INEQUALITIES FOR POISSON INTEGRALS WITH SLOWLY GROWING DIMENSIONAL CONSTANTS

Let Pt be the Poisson kernel. We study the following Lp inequality for the Poisson integral Pf(x, t) = (Pt * f)(x) with respect to a Carleson measure μ: ${\left\| {pf} \right\|_{{L^{p`}}\left( {\mathbb{R}_ + ^{n + 1},du} \right)}} \leqslant {c_{p,n}}k{\left( \mu \right)^{\frac{1}{p}}}{\left\| f \right\|_{{L^p}\left( {{\mathbb{R}^n},dx} \right)}}$ where 1 < p < ∞ and k(μ) is the Carleson norm of μ. It was shown by Verbitsky [V] that for p > 2 the constant cp,n can be taken to be independent of the dimension n. We show that ${c_{2,n}} = O\left( {{{\left( {\log n} \right)}^{\frac{1}{2}}} \right)$ and that ${c_{2,n}} = O\left( {{n^{\frac{1}{p} - \frac{1}{2}}} \right)$ for 1 < p < 2 as n → ∞. We observe that standard proofs of this inequality rely on doubling properties of cubes and lead to a value of Cp,n that grows exponentially with n.