On the $$A_{\infty }$$ A ∞ conditions for general bases

Abstract

We discuss several characterizations of the A ∞ class of weights in the setting of general bases. Although they are equivalent for the usual Muckenhoupt weights, we show that they can give rise to different classes of weights for other bases. We also obtain new characterizations for the usual A ∞ weights.

The Calderón operator and the Stieltjes transform on variable Lebesgue spaces with weights

“…In this paper, our goal is to generalize these results in two ways. First, we extend the approach in [19] to give a new proof of Theorem 1.1 using the maximal operator N . We will do so using a Hedberg type inequality [28].…”

“…If we let C = max{C 1 , C 2 } we get the desired inequality. We now want to define a condition analogous to the A ∞ condition but associated with the basis of intervals {(0, b)} b>0 (as considered in [19]). Hereafter, given an exponent p(·) and a weight w, we define the weight W (x) = w(x) p(x) and denote W (E) = E W (x) dx for any measurable set E ⊂ (0, ∞).…”

Improved Buckley’s theorem on locally compact abelian groups

“…The family of A p classes is increasing and this motivates the definition of the larger class A ∞ as the union A ∞ = p≥1 A p . There are many characterizations of the class A ∞ (see [DMRO16] or the more classical reference [Gra04]). Some of them are given in terms of the finiteness of some A ∞ constant suitably defined.…”

General maximal operators and the reverse Hölder classes

“…However, these characterizations are not valid for a general basis B instead of a basis Q. In the recent paper [5], Duoandikoetxea, Martín-Reyes and Ombrosi compared several characterizations of A ∞,B on a σ-finite measure space (X, µ), a basis is a collection of µ-measurable subsets B of X such that 0 < µ(B) < ∞. They established several implications among such conditions without further assumptions on the basis (or, for example, assuming the boundedness of the maximal operator associated with B), but their assumptions could not decide whether the weights w ∈ A ∞,B belong to RH 1,B .…”