Asymptotical stability of Muckenhoupt weights through Gurov-Reshetnyak classes (Q2844854)

The Gurov-Reshetnyak class of weights with respect to a doubling measure \(\mu\), \(GR_{\varepsilon}(\mu)\), \(0<\varepsilon<2\), was introduced in 1975 as the class of weights for which the following inequality holds NEWLINE\[NEWLINE\frac{1}{\mu(B)} \int_B |w-w_B| \,d\mu \leq \varepsilon w_BNEWLINE\]NEWLINE for all balls \(B\subseteq X\).NEWLINENEWLINEThe main result of the paper shows that weights in the class \(GR_{\varepsilon}(\mu)\) satisfy a weak-reverse Hölder inequality: there exists \(p(\varepsilon)>1\) such that whenever \(1\leq p< p(\varepsilon)\), NEWLINE\[NEWLINE \left(\frac{1}{\mu(B)} \int_B w^p \, d\mu \right)^{1/p} \leq C \frac{1}{\mu(2B)} \int_{2B} w \, d\mu,NEWLINE\]NEWLINE for all balls \(B\) and with a constant \(C\) depending on \(\varepsilon\) and on the doubling constant of the measure \(c_{\mu}\).NEWLINENEWLINEAs an application, some results concerning the embeddings between Muckenhoupt classes \(A_p(\mu)\) and reverse Hölder classes are obtained and, more exactly, the asymptotical behaviour of the \(A_1(\mu)\) condition as the constant \(c_{w}\) in the definition of the \(A_p(\mu)\) class tends to one.