A new formula for the L-p norm

Recently, Brezis, Van Schaftingen and the second author [4] established a new formula for the. W-1,W-p norm of a function in C-c(infinity) (R-N). The formula was obtained by replacing the L-p(R-2N) norm in the Gagliardo semi-norm for. W-s,W-p(R-N) with a weak-L-p(R-2N) quasi-norm and setting s = 1. This provides a characterization of such. W-1,W- p norms, which complements the celebrated Bourgain-Brezis-Mironescu (BBM) formula [1]. In this paper, we obtain an analog for the case s = 0. In particular, we present a new formula for the Lpnorm of any function in L-p(R-N), which involves only the measures of suitable level sets, but no integration. This provides a characterization of the norm on L-p(R-N), which complements a formula by Mazya and Shaposhnikova [12]. As a result, by interpolation, we obtain a new embedding of the Triebel-Lizorkin space F-2(s,p)(R-N)(i.e. the Bessel potential space (I - Delta)(-s/2) L-p(R-N)), as well as its homogeneous counterpart. F-2(s,p)(R-N), for s is an element of(0, 1), p is an element of(1, infinity). (C) 2021 Elsevier Inc. All rights reserved.