Inequalities for Poisson integrals with slowly growing dimensional constants

• Autores: Loukas Grafakos , Enrico Laeng, Carlo Morpurgo
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 51, Nº 1, 2007, págs. 59-75
• Idioma: inglés
• DOI: 10.5565/publmat_51107_04
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• Resumen

  • Let Pt be the Poisson kernel. We study the following Lp inequality for the Poisson integral P f(x, t) = (Pt ∗ f)(x) with respect to a Carleson measure µ: ||P f||Lp(R n+1 + ,dµ) ≤ cp,nκ(µ) 1 p ||f||Lp(Rn,dx) , where 1 < p < ∞ and κ(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 c2,n = O((log n) 1 2 ) and that cp,n = O(n 1 p − 1 2 ) 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.