Duality for $A_{\infty }$ weights on the real line

Abstract

Under the same bounds on $G_q$-constants and $A_p$-constants, the optimal exponents for sharp inclusions between Gehring $G_q$-class of weights and Muckenhoupt $A_p$-class ($1<p,q<\infty$) are Hölder conjugate, if $p$ and $q$ are conjugate. This is a consequence of a representation theorem of $A_{\infty }$ weights in terms of $W^{1,r}$-biSobolev maps and a duality result between $G_q$ and $A_p$ classes in dimension one. We prove also that sharp a priori bounds on constants correspond under the Hölder conjugate mapping $\varphi (t)=\frac{t}{t-1}$.