Characterizations of Sobolev spaces via averages on balls

Abstract

In this paper, the authors prove several equivalent characterizations of Sobolev spaces of even integer orders on Rn, using the average Btf(x)≔1|B(x,t)|∫B(x,t)f(y)dy of a function f over the ball B(x,t)≔{y∈Rn:|y−x|<t} with x∈Rn and t∈(0,∞). These characterizations rely only on the metric and the Lebesgue measure on Rn and are simpler than those obtained recently by Alabern et al. (2012). Moreover, these results may shed new light on the theory of high order Sobolev spaces on spaces of homogeneous type.