A new formula for the $L^p$ norm

Abstract

Recently, Brezis, Van Schaftingen and the second author established a new formula for the ${\dot{W}}^{1,p}$ norm of a function in $C_c^{\infty }({\mathbb{R}}^N)$. The formula was obtained by replacing the $L^p({\mathbb{R}}^{2N})$ norm in the Gagliardo semi-norm for ${\dot{W}}^{s,p}({\mathbb{R}}^N)$ with a weak-$L^p({\mathbb{R}}^{2N})$ quasi-norm and setting $s=1$. This provides a characterization of such ${\dot{W}}^{1,p}$ norms, which complements the celebrated Bourgain-Brezis-Mironescu (BBM) formula. In this paper, we obtain an analog for the case $s=0$. In particular, we present a new formula for the $L^p$ norm of any function in $L^p({\mathbb{R}}^N)$, which involves only the measures of suitable level sets, but no integration. This provides a characterization of the norm on $L^p({\mathbb{R}}^N)$, which complements a formula by Maz'ya and Shaposhnikova. As a result, by interpolation, we obtain a new embedding of the Triebel-Lizorkin space $F_{p,2}^s({\mathbb{R}}^N)$ (i.e. the Bessel potential space $(I-\Delta {)}^{-s/2}{L}^p({\mathbb{R}}^N)$), as well as its homogeneous counterpart ${\dot{F}}_{p,2}^s({\mathbb{R}}^N)$, for $s\in (0,1)$, $p\in (1,\infty )$.

17 pages, updated discussion of Theorem 1.3, and dedication