×

A dimension-free Carleson measure inequality. (English) Zbl 0959.31006

Havin, V. P. (ed.) et al., Complex analysis, operators, and related topics. The S. A. Vinogradov memorial volume. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 113, 393-398 (2000).
Let \(f\in L^p(\mathbb{R}^n)\), where \(1< p< \infty\), and let \(P[f]\) denote the Poisson integral of \(f\) in the halfspace \(D= \mathbb{R}^n\times (0,\infty)\). Further, let \(\mu\) be a Borel measure on \(D\) and let \(\kappa[\mu]= \sup_B\mu(\widehat B)/|B|\), where \(B\subset \mathbb{R}^n\) is an open ball, \(|B|\) is its volume, \(\widehat B= B\times (0,r)\) and \(r\) is the radius of \(B\). The Carleson measure inequality states that \(\|P[f]\|_{L^p(\mu)}\leq C(n, p)(\kappa[\mu])^{1/p}\|f\|_p\). Some time ago S. A. Vinogradov asked whether this inequality is true with a constant \(C(p)\) independent of \(n\). The author shows that this is the case when \(p> 2\). The case where \(1< p\leq 2\) is left open.
For the entire collection see [Zbl 0934.00031].

MSC:

31B10 Integral representations, integral operators, integral equations methods in higher dimensions
42B25 Maximal functions, Littlewood-Paley theory
PDFBibTeX XMLCite