Bi-Sobolev homeomorphisms and Bpq classes

Let (Formula presented.) be a bi-Sobolev map, that is a homeomorphism which is locally absolutely continuous with its inverse and let (Formula presented.) ((Formula presented.)) be the class of weights w on (Formula presented.) satisfying the reverse Hölder inequality [Equation not available: see fulltext.]for every interval (Formula presented.). Assume (Formula presented.) belongs to a (Formula presented.) class. We look for conditions under which also (Formula presented.) belongs to some related (Formula presented.) class. Namely we prove that, if (Formula presented.) and (Formula presented.), then (Formula presented.)Moreover we show some links among (Formula presented.) classes and Muckenhoupt and Gehring classes and we compute the (Formula presented.)-constant of a power function.