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The sharp \(A_p\) constant for weights in a reverse-Hölder class. (English) Zbl 1174.42024

R. R. Coifman and C. Fefferman [Stud. Math. 51, 241–250 (1974; Zbl 0291.44007)] established that the class of Muckenhoupt \(A_p\) weights is equivalent to the class of reverse-Hölder \(RH_p\) weights. Let \(1<p<\infty\). Fix an interval \(I\) and define \[ \begin{aligned} [w]_{A_p(I)} &= \sup_{J \subset I} \Big( \frac{1}{| J |} \int_J w(x)dx \Big) \Big( \frac{1}{| J |} \int_J w(x)^{1-p'}dx \Big)^{p-1}, \\ [w]_{RH_p(I)} &= \sup_{J \subset I} \frac{ \big( \frac{1}{| J |} \int_J w(x)^pdx \big)^{1/p} } { \frac{1}{| J |} \int_J w(x)dx }. \end{aligned} \] V. I. Vasyunin [St. Petersbg. Math. J. 15, No. 1, 49–79 (2004; Zbl 1057.42017)] found the sharp constant \(C_{p,q,\delta}\) such that \[ \text{If} \quad [w]_{A_p(I)} \leq \delta \quad \text{then} \quad [w]_{RH_q(I)} \leq C_{p,q,\delta}. \] The authors prove the inverse, that is, they find the sharp constant \(C_{p,q,\delta}'\) such that \[ \text{If} \quad [w]_{RH_p(I)} \leq \delta \quad \text{then} \quad [w]_{A_q(I)} \leq C_{p,q,\delta}'. \] To prove this, they use the Bellman function technique. They also consider it on \(\mathbb{R}^n\), without claiming the sharpness.

MSC:

42B30 \(H^p\)-spaces
42B25 Maximal functions, Littlewood-Paley theory
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References:

[1] Basile, L., D’Apuzzo, L. and Squillante, M.: The limit class of Gehring type \(G_\infty\). Boll. Un. Mat. Ital. B (7) 11 (1997), no. 4, 871-884. · Zbl 0941.42009
[2] Basile, L., D’Apuzzo, L. and Squillante, M.: The limit class of Gehring type \(G_\infty\) in the \(n\)-dimensional case. Rend. Mat. Appl. (7) 21 (2001), no. 1-4, 207-221. · Zbl 1042.42012
[3] Coifman, R.R. and Fefferman, C.: Weighted norm inequalities for maximal functions and singular integrals. Studia Math. 51 (1974), no. 3, 241-250. · Zbl 0291.44007
[4] Cruz-Uribe SFO, D. and Neugebauer, C.J.: The structure of the reverse Hölder classes. Trans. Amer. Math. Soc. 347 (1995), no. 8, 2941-2960. JSTOR: · Zbl 0851.42016 · doi:10.2307/2154763
[5] Fefferman, R., Kenig, C. and Pipher, J.: The theory of weights and the Dirichlet problem for elliptic equations. Ann. of Math. (2) 134 (1991), no. 1, 65-124. JSTOR: · Zbl 0770.35014 · doi:10.2307/2944333
[6] García-Cuerva, J. and Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics . North-Holland Mathematics Studies 116 . North-Holland Publishing Co., Amsterdam, 1985. · Zbl 0578.46046
[7] Gehring, F.W.: The \(L_p\)-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130 (1973), 265-277. · Zbl 0258.30021 · doi:10.1007/BF02392268
[8] Hruščev, S.: A description of weights satisfying the \(A_\infty\) condition of Muckenhoupt. Proc. Amer. Math. Soc. 90 (1984), 253-257. JSTOR: · Zbl 0539.42009 · doi:10.2307/2045350
[9] Kinnunen, J.: Sharp results on reverse Hölder inequalities. Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes No. 95 , 1994, 34 pp. · Zbl 0816.26008
[10] Korenovskii, A.A.: Sharp extension of a reverse Hölder inequality and the Muckenhoupt condition (Russian). Mat. Zametki 52 (1992), no. 6, 32-44, 158; translation in Math. Notes 52 (1992), no. 5-6, 1192-1201.
[11] Malaksiano, N.A.: On exact inclusions of Gehring classes in Muckenhoupt classes (Russian). Mat. Zametki 70 (2001), no. 5, 742-750; translation in Math. Notes 70 (2001), no. 5-6, 673-681. · Zbl 1024.30018 · doi:10.1023/A:1012983028054
[12] Muckenhoupt, B.: The equivalence of two conditions for weight functions. Studia Math. 49 (1973/74), 101-106. · Zbl 0243.44003
[13] Nazarov, F., Treil, S. and Volberg, A.: Bellman function in stochastic control and harmonic analysis. In Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000) , 393-423. Oper. Theory Adv. Appl. 129 . Birkhäuser, Basel, 2001. · Zbl 0999.60064
[14] Neugebauer, C.J.: The precise range of indices for the \(RH_R\)- and \(A_P\)-weight classes. Preprint,
[15] Rios, C.: \(L^p\) regularity of the Dirichlet problem for elliptic equations with singular drift. Publ. Mat. 50 (2006), no. 2, 475-507. · Zbl 1192.35184 · doi:10.5565/PUBLMAT_50206_11
[16] Stein, E.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals . Princeton Mathematical Series 43 . Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993. · Zbl 0821.42001
[17] Vasyunin, V.: The exact constant in the inverse Hölder inequality for Muckenhoupt weights. St. Petersburg Math. J. 15 (2004), no. 1, 49-79. · Zbl 1057.42017 · doi:10.1090/S1061-0022-03-00802-1
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