Abstract
Yano's extrapolation theorem dated back to 1951 establishes boundedness properties of a subadditive operator acting continuously in for close to and/or taking into as and/or with norms blowing up at speed and/or,. Here we give answers in terms of Zygmund, Lorentz-Zygmund and small Lebesgue spaces to what happens if as. The study has been motivated by current investigations of convolution maximal functions in stochastic analysis, where the problem occurs for . We also touch the problem of comparison of results in various scales of spaces.
Highlights
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
We give answers in terms of Zygmund, Lorentz-Zygmund and small Lebesgue spaces to what happens if T f 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
Summary
Throughout the paper Ω ⊂ RN will be measurable and with Lebesgue measure |Ω| = 1. The latter is purely technical, any |Ω| < ∞ can be considered. If f is a (real) measurable function on Ω, we will use the standard symbol f ∗ for its nonincreasing rearrangement— see, for example, [2, 10]. The usual Lebesgue space of functions integrable with the pth power will be denoted by Lp = Lp(Ω); we will use the averaging norm f p=. Should no misunderstanding occur we will sometimes denote various constants in formulas by the same symbol.
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