Characterizations of Sobolev spaces via averages on balls

In this paper, the authors prove several equivalent characterizations of Sobolev spaces of even integer orders on RnRn, using the average Btf(x)≔1|B(x,t)|∫B(x,t)f(y)dy of a function ff over the ball B(x,t)≔{y∈Rn:|y−x|