The limit class of Gehring type Ginf/ty in the n-dimensional case

In the paper [BSW] a link between the Muckenhoupt class Al and the Gehring classes Gq was established. More exactly; denoting by I the interval (0,1), the following theorem holds, THEOREM [BSW] (see Corollary I and Theorem 2). -Suppose that f is a function in Al (1; c). Then, for every' p <C I(c -1), (a) fEGp(I;C p), cp "' .. -I1.. ; cp (c+p-cp) (b) jPEA1(I;c p), £p= __c -c+p-cp The constant cp and fp cannatbe improved. In the present paper we get similar relations in the case of the limit class Goo; in particular we prove that: ifjis in Goo{I;r), therr.for every p> c,j is in Ap(I; (1 I c)(p -1) I(p -c»p -I) and jI/(p-ll is in Gx(I;(p-l)/(p-c»); moreover the constant (ljc)«p-l)/(pr))P -1 is optimal. Finally we' stress that 3. certain duality results from the preceding relations.