JournalsrmiVol. 30, No. 4pp. 1191–1236

Equivalent definitions of dyadic Muckenhoupt and reverse Hölder classes in terms of Carleson sequences, weak classes, and comparability of dyadic LL log LL and AA_\infty constants

  • Oleksandra Beznosova

    University of Alabama, Tuscaloosa, USA
  • Alexander Reznikov

    Vanderbilt University, Nashville, USA
Equivalent definitions of dyadic Muckenhoupt and reverse Hölder classes in terms of Carleson sequences, weak classes, and comparability of dyadic $L$ log $L$ and $A_\infty$ constants cover
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Abstract

In the dyadic case the union of the reverse Hölder classes, p>1RHpd\cup_{p>1} RH_p^d, is strictly larger than the union of the Muckenhoupt classes, p>1Apd=Ad\cup_{p>1} A_p^d = A_\infty^d. We introduce the RH1dRH_1^d condition as a limiting case of the RHpdRH_p^d inequalities as pp tends to 11 and show the sharp bound on the RH1dRH_1^d constant of the weight ww in terms of its AdA_\infty^d constant.

We also examine the summation conditions of the Buckley type for the dyadic reverse Hölder and Muckenhoupt weights and deduce them from an intrinsic lemma which gives a summation representation of the bumped average of a weight. We also obtain summation conditions for continuous reverse Hölder and Muckenhoupt classes of weights and both continuous and dyadic weak reverse Hölder classes. In particular, we prove that a weight belongs to the class RH1RH_1 if and only if it satisfies Buckley's inequality. We also show that the constant in each summation inequality of Buckley type is comparable to the corresponding Muckenhoupt or reverse Hölder constant. To prove our main results we use the Bellman function technique.

Cite this article

Oleksandra Beznosova, Alexander Reznikov, Equivalent definitions of dyadic Muckenhoupt and reverse Hölder classes in terms of Carleson sequences, weak classes, and comparability of dyadic LL log LL and AA_\infty constants. Rev. Mat. Iberoam. 30 (2014), no. 4, pp. 1191–1236

DOI 10.4171/RMI/812