A new formula for the L norm

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