Sharp interactions among A_infty -weights on the real line

摘要

We prove that any weight (vin L^1_{loc}({{mathbb {R}}})) (v:{{mathbb {R}}}rightarrow [0,+infty )) of the Muckenhoupt class (A_infty ) has the form (v=h^prime ) where (h:{{mathbb {R}}}rightarrow {{mathbb {R}}}) is a bi-Sobolev map. As an application we improve the known results on exact continuation and reciprocal imbeddings for Gehring (G_q) and Muckenhoupt (A_p) classes, providing exact bounds in all cases. The method relies on a duality formula due to Johnson and Neugebauer [Rev Mat Iberoam 3(2)249–273, (1987)] $$begin{aligned} A_p((h^{-1} )^prime ) =G_q(h^prime ), end{aligned}$$ (frac{1}{p}+frac{1}{q}=1). Keywords Muckenhoupt weights Gehring classes Bi-Sobolev maps Mathematics Subject Classification 26A46 26A12 46E30 Communicated by Salvatore Rionero.