We are concerned with the problem of finding sharp summability conditions on the wvights which render certain weighted inequalities of PoincarB-type true. The conditions we find ixiitsist of proper integral balances between the growths of the rearrangements of the weights.
I . Introduction
We deal with weighted PoincarC inequalities, a prototype of which amounts to the li )]lowing statement:
for any (sufficiently smooth) realvalued function u on G such that u = 0 on a G . l l i w , and in what follows, G is an open subset of IRn having Lebesgue measure IGl;
ilnd v are locally integrable nonnegative functions; p and q belong to (1, + 00); C clitiiotes a constant which may depend on G, v, w , p , q but not on u.
Inequalities of type (1.1) and related ones have been studied especially in view of ri1)plications to degenerate partial differential equations (see e. g. [MS], [FKS]). A c,hriracterisation of those weights which render (1.1) satisfied is given, for w = v, IJY the so-called Muckenhoupt condition [MW]. An exhaustive treatment of weighted I'oincarC inequalities in terms of the notion of capacity is presented in (MI. In [EE] and IM)] the study of such inequalities has been linked to a measure of non-compactness I991 Mathematica Subject Classificntion. Primary 26 D 10. Keywords and phmses. PoincarB inequalities, weights, rearrangements, isoperimetric function.
Centre for Maihemaiical Analysis and i i s Applications Czech Universiiy University of Sussez of Agriculiure Falmer