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

Ziyi He, Dachun Yang and Wen Yuan

Mathematische Nachrichten, 2018, vol. 291, issue 2-3, 284-325

Abstract: 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].

Date: 2018
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