The sharp ApA_p constant for weights in a reverse-Hölder class

  • Martin Dindoš

    Edinburgh University, UK
  • Treven Wall

    Edinburgh University, UK


Coifman and Fefferman established that the class of Muckenhoupt weights is equivalent to the class of weights satisfying the "reverse Hölder inequality". In a recent paper V. Vasyunin [The exact constant in the inverse Hölder inequality for Muckenhoupt weights. St. Petersburg Math. J. 15 (2004), no. 1, 49-79] presented a proof of the reverse Hölder inequality with sharp constants for the weights satisfying the usual Muckenhoupt condition. In this paper we present the inverse, that is, we use the Bellman function technique to find the sharp ApA_p constants for weights in a reverse-Hölder class on an interval; we also find the sharp constants for the higher-integrability result of Gehring [The LpL_p-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130 (1973), 265-277]. Additionally, we find sharp bounds for the ApA_p constants of reverse-Hölder-class weights defined on rectangles in Rn\mathbb{R}^n, as well as bounds on the ApA_p constants for reverse-Hölder weights defined on cubes in Rn\mathbb{R}^n, without claiming the sharpness.

Cite this article

Martin Dindoš, Treven Wall, The sharp ApA_p constant for weights in a reverse-Hölder class. Rev. Mat. Iberoam. 25 (2009), no. 2, pp. 559–594

DOI 10.4171/RMI/576