Abstract
A classical fact in the weighted theory asserts that a weight w belongs to the Muckenhoupt class A_infty if and only if its logarithm log w is a function of bounded mean oscillation. We prove a sharp quantitative version of this fact in dimension one: for a weight w defined on some interval Jsubset mathbb {R}, we provide best lower and upper bounds for the BMO norm of log w in terms of A_infty characteristics of w. The proof rests on the precise evaluation of associated Bellman functions.
Highlights
A real-valued locally integrable function φ defined on Rn is said to be in B M O, the space of functions of bounded mean oscillation, if sup |φ − φ Q| Q < ∞
Due to Fefferman [4], asserts that B M O is a dual to the Hardy space H 1
It is well-known that the functions of bounded mean oscillation have very strong integrability properties
Summary
A real-valued locally integrable function φ defined on Rn is said to be in B M O, the space of functions of bounded mean oscillation, if sup |φ − φ Q| Q < ∞. Following [8], we say that a weight w satisfies the condition A∞ (or belongs to the class A∞(B)), if [w]A∞(B) = sup w Q exp(− log w Q) < ∞, where, as above, the supremum is taken over all cubes Q contained in the base space B, having edges parallel to the coordinate axes. Our approach will rest on the so-called Bellman function method, a powerful technique which is used widely in various contexts of analysis and probability theory Speaking, this technique enables to extract the optimal constants in a given estimate from the existence of a certain special function enjoying an appropriate size condition and concavity. The formal verification that these guessed objects are the desired Bellman functions, is the contents of Sect. 3
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