On the embedding of 𝐴₁ into 𝐴_{∞}

Abstract

We give a quantitative embedding of the Muckenhoupt class A 1 A_1 into A ∞ A_\infty . In particular, we show how ϵ \epsilon depends on [ w ] A 1 [w]_{A_1} in the inequality which characterizes A ∞ A_\infty weights: \[ w ( E ) w ( Q ) ≤ ( | E | | Q | ) ϵ , \frac {w(E)}{w(Q)} \leq \biggl ( \frac {|E|}{|Q|} \biggr )^\epsilon , \] where Q Q is any dyadic cube and E E is any subset of Q Q . This embedding yields a sharp reverse-Hölder inequality as an easy corollary.

Bellman function for extremal problems in BMO

“…The corresponding sharp weak-type Reverse Hölder inequalities for A p1,p2 -weights were established in [36]; the latter paper uses the Bellman function for f(x) = χ [1,∞) (x 1 ). See [35] for the limiting case of the Muckenhoupt class A 1 .…”

Asymptotically sharp reverse Hoelder inequalities for flat Muckenhoupt weights

“…As a corollary we get some useful sharp estimates for A 1 and A ∞ weights. These should be compared to the results and discussion in [25].…”

“…Indeed, such endpoint estimates are proved for example in [22] in the case that w is a dyadic A 1 weight. See also [25] for a Bellman function approach. We provide here the analog of the result in [22] for non-dyadic weights in A 1 , together with the corresponding endpoint reverse Hölder inequality for A ∞ weights on the real line.…”

“…Indeed, such endpoint estimates are proved for example in [22] in the case that w is a dyadic A 1 weight. See also [25] for a Bellman function approach.…”