Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 74960, 15 pages
doi:10.1155/JIA/2006/74960
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.