Bi-Sobolev homeomorphisms and Bqp classes

Let h : IR -> IR be a bi-Sobolev map, that is a homeomorphism which is locally absolutely continuous with its inverse and let B-p(q) (p < q) be the class of weights w on IR satisfying the reverse Holder inequality (f(I) w(q) (x)dx)(1/q) <= K (f(I) w(q) (x)dx)(1/p) , (K > 1), for every interval I. IR. Assumew = h' belongs to a B-p(q) class. We look for conditions under which also v = (h(-1))' belongs to some related B-r(s) class. Namely we prove that, if p < 0 and q > 1, then (h(-1))' epsilon B-1-q(1-p) double left right arrow h' epsilon B-p(q). Moreover we show some links among B-r(s) classes and Muckenhoupt and Gehring classes and we compute the B-r(s)-constant of a power function.