Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 74960, 15 pages
doi:10.1155/JIA/2006/74960
On extrapolation blowups in the LP scale
1CNR Istituto per le Applicazioni del Calcolo “Mauro Picone”, Via P. Castellino 111, Napoli 80131, Italy
2Dipartimento di Costruzioni e Metodi Matematici in Architettura, Università degli Studi di Napoli “ Federico II”, via Monteoliveto 3, Napoli 80134, Italy
3CNR Istituto per le Applicazioni del Calcolo “Mauro Picone”, Via P. Castellino 111, Napoli 80131, Italy
4Institute of Mathematics, Academy of Sciences of the Czech Republic, Žitná 25, Prague 1 CZ-115 67, Czech Republic
Received 15 October 2004; Revised 31 March 2005; Accepted 6 April 2005
Copyright © 2006 Claudia Capone et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Yano's extrapolation theorem dated back to 1951 establishes boundedness properties of a subadditive operator T acting continuously in Lp for p close to 1
and/or taking L∞ into Lp as p→1+ and/or p→∞ with norms blowing up at speed
(p−1)−α and/or pβ, α,β>0. Here we give answers in terms of Zygmund, Lorentz-Zygmund and small Lebesgue spaces to what happens if ‖Tf‖p≤c(p−r)−α‖f‖p as p→r+(1<r<∞). The study has been motivated by current investigations of convolution maximal functions in stochastic analysis, where the problem occurs for r=2 . We also touch the problem of comparison of results in various scales of spaces.