Littlewood–Paley characterizations of higher†order Sobolev spaces via averages on balls

In this article, the authors characterize higher†order Sobolev spaces Wα,p(Rn), with n∈[4,∞), α∈2N and p∈(2nn+4,∞), or with n∈N, α∈(0,∞)∖2N and p∈(max{2nn+2(α−2⌊α/2⌋),1},∞), via the Lusin area function and the Littlewood–Paley gλ*†function in terms of ball averages, where ⌊α/2⌋ denotes the maximal integer not greater than α/2. Moreover, the authors also complement the above results in the endpoint cases of p via establishing some weak type estimates. These improve and develop the corresponding known results for Sobolev spaces with smoothness order α∈(0,2].