Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2021-04-23 , DOI: 10.1016/j.jfa.2021.109075 Qingsong Gu , Po-Lam Yung
Recently, Brezis, Van Schaftingen and the second author [4] established a new formula for the norm of a function in . The formula was obtained by replacing the norm in the Gagliardo semi-norm for with a weak- quasi-norm and setting . This provides a characterization of such norms, which complements the celebrated Bourgain-Brezis-Mironescu (BBM) formula [1]. In this paper, we obtain an analog for the case . In particular, we present a new formula for the norm of any function in , which involves only the measures of suitable level sets, but no integration. This provides a characterization of the norm on , which complements a formula by Maz′ya and Shaposhnikova [12]. As a result, by interpolation, we obtain a new embedding of the Triebel-Lizorkin space (i.e. the Bessel potential space ), as well as its homogeneous counterpart , for , .
中文翻译:
L p范数的新公式
最近,Brezis,Van Schaftingen和第二作者[4]建立了新的公式 功能的规范 。该公式是通过替换 Gagliardo半范数中的范数 用弱 准规范和设定 。这提供了这样的特征规范,这是对著名的布尔加斯-布列兹-米罗涅斯库(BBM)公式的补充[1]。在本文中,我们获得了一个类似的案例。特别是,我们为 在任何功能的规范 ,它仅涉及适当级别集的度量,而没有积分。这提供了规范上的特征,这是对Maz'ya和Shaposhnikova [12]公式的补充。结果,通过插值,我们获得了Triebel-Lizorkin空间的新嵌入 (即贝塞尔潜在空间 ),以及同类同类产品 , 为了 , 。