Inequalities for Poisson integrals with slowly growing dimensional constants (Q2454910)

For \(1<p<\infty\) let \(Pf (x,t)\) denote the Poisson integral of \(f \in L^p (\mathbb R^n)\). The main object of study of this paper is the inequality \[ \| Pf\| _{L^p (\mathbb R^{n+1}_+; \, d\mu)}\leq c_{p,n}\, \kappa (\mu)^{1/p}\,\| f \| _{L^p (\mathbb R^{n})}, \] where \(d \mu\) is the Carleson measure in the half-space \(\mathbb R^{n+1}_+\), and \(\kappa (\mu)\) is the Carleson norm of \(\mu\). For \(p>2\) this inequality was proved by \textit{I. E. Verbitsky} [in: Complex analysis, operators, and related topics. The S. A. Vinogradov memorial volume. Oper. Theory, Adv. Appl. 113, 393--398 (2000; Zbl 0959.31006)], showing that the constant \(c_{p,n}\) can be chosen independent of \(n\). The author extend these result, proving the inequality above for \(1<p<2\) with \(c_{p,n}=C_p \, n^{1/p -1/2}\), as well as for \(p=2\) with \(c_{2,n}=C\, (\log n)^{1/2}\). These estimates of the constant \(c_{p,n}\) improve the previously known estimates.