The sharp constant in the reverse Hölder inequality for Muckenhoupt weights

Abstract

Abstract. Coifman and Fefferman proved that the "reverse Hölder inequality" is fulfilled for any weight satisfying the Muckenhoupt condition. In order to illustrate the power of the Bellman function technique, Nazarov, Volberg, and Treil showed (among other things) how this technique leads to the reverse Hölder inequality for the weights satisfying the dyadic Muckenhoupt condition on the real line. In this paper the proof of the reverse Hölder inequality with sharp constants is presented for the weights satisfying the usual (rather than dyadic) Muckenhoupt condition on the line. The results are a consequence of the calculation of the true Bellman function for the corresponding extremal problem.In [1] Coifman and Fefferman proved that for any weight (i.e., for any nonnegative function) w on R n satisfying the Muckenhoupt condition(the supremum is taken over all cubes Q with edges parallel to the coordinate axes) the "reverse Hölder inequality" is fulfilled:for some q > 1, with a constant C independent of the cube Q. Here and later, ϕ Q stands for the average of the function ϕ over the set Q:In [2] Nazarov, Treil, and Volberg illustrated the power of the Bellman function technique. In particular, they deduced the reverse Hölder inequality for dyadic A ∞ -weights on the real line by constructing a Bellman function. In the present paper we refine this technique to prove the reverse Hölder inequality for arbitrary A p -weights (1 ≤ p ≤ ∞) on the real line with a sharp constant C depending on p and q (the exponent in the reverse Hölder inequality), and on the A p -"norm" δ of w. Furthermore, we find the sharp bound for the exponents q for which the reverse Hölder inequality is true. Instead of Muckenhoupt's original definition of A ∞ -weights (used, e.g., in [1]), we employ the equivalent definition introduced by Khrushchëv [3]. Namely, the symbol A δ ∞ (J) will denote the set of all nonnegative functions w ∈ L 1 (J) such that (1) sup I⊂J w I exp(− log w I ) ≤ δ < ∞.2000 Mathematics Subject Classification. Primary 42B20, 42B25.

Bellman function for extremal problems in BMO

“…where Q > 1, meaning [w] ∞ Q whenever ϕ = (log w, w) belongs to A Ω 1,∞,Q . This domain and the study of the corresponding Bellman functions go back to [53]. See [1] for the study of this particular case and applications of the obtained inequalities.…”

“…The paper [54] generalizes [53], where the first sharp Bellman functions for Muckenhoupt weights were found; those Bellman functions lead to sharp constants in the Reverse Hölder inequality. The scale A p1,p2 includes the classical Muckenhoupt classes A p .…”

The Gehring Lemma: Dimension free estimates

“…where now R is any rectangle in R n (no bound on the ratios between side-lengths). See also [6,21,24,30,32] for further interesting results in this direction. In the present note we show that dimension free integrability improvement follows from an inhomogeneous weak reverse Hölder inequality on rectangles R whose side lengths have ratios between 1/2 and 2 (we call them admissible rectangles):…”

Weighted Solyanik estimates for the strong maximal function

“…Sharp quantitative versions of these inequalities are contained in several places in the literature as for example in [19,20] and [23] for the one-parameter case, and in [22,27] for the multiparameter case. In one dimension even more precise results are known which also describe the optimal numerical constants involved in the estimates; see for example [5] and [32]. As a corollary of Theorem 1.3 we obtain a reverse Hölder inequality for strong Muckenhoupt weights.…”