- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:45
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
25
2009
2
The sharp $A_p$ constant for weights in a reverse-Hölder class
Martin
Dindoš
Edinburgh University, EDINBURGH, UNITED KINGDOM
Treven
Wall
Edinburgh University, EDINBURGH, UNITED KINGDOM
Reverse-Hölder class, Gehring class, $A_p$ weight, Muckenhoupt weight, Bellman function
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 $A_p$ 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 $L_p$-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130 (1973), 265-277]. Additionally, we find sharp bounds for the $A_p$ constants of reverse-Hölder-class weights defined on rectangles in $\mathbb{R}^n$, as well as bounds on the $A_p$ constants for reverse-Hölder weights defined on cubes in $\mathbb{R}^n$, without claiming the sharpness.
Fourier analysis
General
559
594
10.4171/RMI/576
http://www.ems-ph.org/doi/10.4171/RMI/576