Weighted Sobolev Interpolation Inequalities on Certain Domains

We obtain some weighted Sobolev interpolation inequalities oncertain domains that include Lipschitz domains for doublingweights satisfying a weighted Poincaré inequality. Thesegeneralize most results in Gutierrez and Wheeden's paper [20].We also give some applications on Lipschitz domains for weightsof type dist (·, G)⁸, where G .     

Average value problems in ordinary differential equations

Using Schauder's fixed point theorem, with the help of an integral representation in ‘Sharp conditions for weighted 1-dimensional Poincaré inequalities’, Indiana Univ. Math. J., 49 (2000) 143-175, by Chua and Wheeden, we obtain existence and uniqueness theorems and ‘continuous dependence of average condition’ for average value problem:

y^′=F(x,y),     

Self-improving properties of inequalities of Poincaré type on measure spaces and applications

We show that the self-improving nature of Poincaré estimates persists for domains in rather general measure spaces. We consider both weak type and strong type inequalities, extending techniques of B. Franchi, C. Pérez and R. Wheeden. As an application in spaces of homogeneous type, we derive global Poincaré estimates for a class of domains with rough boundaries that we call ?-John domains, and we show that such domains have the requisite properties. This class includes John (or Boman) domains as well as s-John domains. Further applications appear in a companion paper.