SOME RECENT DEVELOPMENTS OF HIGH ORDER SOBOLEV-TYPE SPACES

作者：Chang, Der-Chen; Yang, Dachun^*; Yuan, Wen; Zhang, Junwei

来源：Journal of Nonlinear and Convex Analysis, 2016, 17(9): 1831-1865.

• Sobolev space
• Besov space
• Triebel-Lizorkin space
• Morrey-Sobolev space
• ball average
• Lusin-area function
• Littlewood-Paley function
• metric measure space
• space of homogeneous type

In this article, we first survey some recent developments of high order Sobolev spaces, namely, Sobolev spaces with smoothness order greater than 1, on the Euclidean space. A series of recently established characterizations of these Sobolev spaces are presented. Some corresponding characterizations for Besov and Triebel-Lizorkin spaces on the Euclidean spaces are also presented in this article. A common advantage of these characterizations lies in that they do not depend on the differential structure of the Euclidean space, and hence provide several possible ways to introduce high order Sobolev, Besov and Triebel-Lizorkin spaces on general metric measure spaces. As another main content of this article, we also establish a series of new pointwise characterizations of Morrey-Sobolev spaces with any positive smoothness order on the Euclidean spaces, in terms of pointwise inequalities involving ball averages, which also serve as possible approaches to introduce high order Morrey-Sobolev spaces on metric measure spaces. Some related open questions about these spaces on metric measure spaces are also presented.

• 出版日期2016
• 单位北京师范大学