On the sum of a Sobolev space and a weighted [formula omitted]-space

Abstract

Let p>n and let Lp1(Rn) be a homogeneous Sobolev space. For an arbitrary Borel measure μ on Rn we give a constructive characterization of the space Σ=Lp1(Rn)+Lp(Rn;μ). We express the norm of this space in terms of certain local oscillations with respect to the measure μ. In particular, we prove that if μ is a measure supported on a finite set S⊂Rn, then there exist a sequence {αi} of positive numbers and a family {xi,yi} of two point subsets of S, i=1,…,m, such that m⩽C(n)#S and for every function f on Rn‖f‖Σp∼∑i=1mαi|f(xi)−f(yi)|p.We also obtain another equivalent expression for the norm of the space Σ which enables us to describe the K-functional for the couple (Lp(Rn;μ),Lp1(Rn)) in terms of p-oscillations of functions, and to prove that this couple is quasi-linearizable.