A Dimension-free Carleson Measure Inequality

Abstract

We consider the following Carleson measure inequality for the Poisson integrals P[f] in ℝ ⁿ⁺¹₊ :

$$\left\| {P[f]} \right\|_{L^p } (\mathbb{R}_ + ^{n + 1} ,d\mu ) \leqslant C\mathcal{H}\left[ \mu \right]^{\frac{1} {p}} \left\| f \right\|L^P (\mathbb{R}^n ),{\text{ }}f \in L^p (\mathbb{R}^n )$$

, where 1 < p < ∞,μ is a Borel measure on ℝ ⁿ⁺¹₊ , and $\kappa [\mu ] = {{\sup }_{B}}\tfrac{{\mu (\hat{B})}}{{|B|}}$ is the Carleson norm of _μ Here $\hat{B} = \{ (x,t):x \in B,0 < t < r\}$ = {(x, t): x ∈ B, 0 < t < r} is a cylinder whose base is the n-ball B of radius r. S. A. Vinogradov asked whether this inequality is fulfilled with a constant C = C _p independent of the dimension n. We prove this inequality for p > 2 and discuss some related results.