A new formula for the Lp norm

Abstract

Recently, Brezis, Van Schaftingen and the second author [4] established a new formula for the W˙1,p norm of a function in Cc∞(RN). The formula was obtained by replacing the Lp(R2N) norm in the Gagliardo semi-norm for W˙s,p(RN) with a weak-Lp(R2N) quasi-norm and setting s=1. This provides a characterization of such W˙1,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 Lp norm of any function in Lp(RN), which involves only the measures of suitable level sets, but no integration. This provides a characterization of the norm on Lp(RN), 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 F2s,p(RN) (i.e. the Bessel potential space (I−Δ)−s/2Lp(RN)), as well as its homogeneous counterpart F˙2s,p(RN), for s∈(0,1), p∈(1,∞).