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 satisf… Show more
“…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.…”
Section: The a P 1 P 2 Classesmentioning
confidence: 99%
“…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 .…”
In this paper we develop the method of finding sharp estimates by using a Bellman function. In such a form the method appears in the proofs of the classical John-Nirenberg inequality and L p estimations of BMO functions. In the present paper we elaborate a method of solving the boundary value problem for the homogeneous Monge-Ampère equation in a parabolic strip for sufficiently smooth boundary conditions. In such a way, we have obtained an algorithm for constructing an exact Bellman function for a large class of integral functionals on the BMO space.
“…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.…”
Section: The a P 1 P 2 Classesmentioning
confidence: 99%
“…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 .…”
In this paper we develop the method of finding sharp estimates by using a Bellman function. In such a form the method appears in the proofs of the classical John-Nirenberg inequality and L p estimations of BMO functions. In the present paper we elaborate a method of solving the boundary value problem for the homogeneous Monge-Ampère equation in a parabolic strip for sufficiently smooth boundary conditions. In such a way, we have obtained an algorithm for constructing an exact Bellman function for a large class of integral functionals on the BMO space.
“…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):…”
Section: Introduction and Statement Of Resultsmentioning
“…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.…”
Let M S denote the strong maximal operator on R n and let w be a non-negative, locally integrable function. For α ∈ (0, 1) we define the weighted sharp Tauberian constant C S,w associated with M S by
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