QUANTITATIVE DEPENDENCE OF SOME DISCRETE LIMITING CLASSES ON THE MUCKENHOUPT [A.sub.1] CLASS

We prove some relations between the discrete Gehring classes [G.sub.q] and discrete Muckenhoupt classes [A.sub.p]. Specifically, by using some known Hardy-type and Carleman-type inequalities, we study the relationship between [G.sub.1], [A.sub.[infinity]], and [A.sub.1] for nonincreasing and nondecreasing weights. Finally, we establish some general results by introducing the notions of [G.sub.[phi]] classes defined for a nonnegative convex function [phi].