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The structure of increasing weights on the real line. (English) Zbl 1177.46019

The structure of a variety of related weight classes on the real line and the positive real axis is examined: doubling measures, \(A_p\) weights, the \(B_p\) weights of Ariño and Muckenhoupt, and \(\Delta_2\) Young functions. A number of characterizations of these classes are given: as applications, the Matuszewska-Orlicz indices of the Young function \(w(t)=t^{a+b\sin(\log(|\log(t)|))}\), \(a,b\in \mathbb{R}^+\), \(a>1+b\sqrt{2}\), due to Linderberg is computed, a sufficient condition for a function \(m\) to be a multiplier of the doubling measures on \(\mathbb{R}^+\) is given, and a function \(W\) on \(\mathbb{R}^+\) is constructed such that \(W(0)=0\), \(W\) is increasing and locally Hölder continuous, but the measure \(\mu\) defined by \(\mu([0,t])=W(t)\) is not a doubling measure.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
26A12 Rate of growth of functions, orders of infinity, slowly varying functions
42B25 Maximal functions, Littlewood-Paley theory

Citations:

Zbl 0716.42016
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