Some remarks on a formula for Sobolev norms due to Brezis, Van Schaftingen and Yung

Abstract

We provide answers to some questions raised in a recent work by H. Brezis, J. 3] concerning the Gagliardo semi-norm |u| W s,q computed at s = 1, when the strong L q is replaced by weak L q . In particular, we address generalization of the results in [2, 3] for a general domain and non-smooth functions. Proof of Theorem 1.5This section is devoted to the proof of Theorem 1.5. Its special case r = q is essential for the proof of the main results Theorem 1.3 and Corollary 1.1. Theorem 1.5 is a particular case of the following more general statment.

A new formula for the L norm

“…allowing us to invoke [11,Lemma 3.1].) This allows us to use the three lines lemma from complex analysis to the bounded holomorphic function H(z)/(z + 1) N +1 on the strip {z ∈ C : 0 ≤ Re z ≤ 1}, and conclude that…”

Going to Lorentz when fractional Sobolev, Gagliardo and Nirenberg estimates fail

“…Proof of Theorem 2 when p = 1. As already mentioned, in this case the conclusion of Theorem 2 is essentially due to A. Poliakovsky [19]. Indeed, if a measurable function u satisfies (1.7), then so does its truncation u h for any h > 0 where u h := max{min{u, h}, −h}.…”

“…The proof of [19, Cor. 1.1] is quite intricate and we refer the reader to [19]; it would be interesting to find a simpler argument as in the case p > 1.…”

Anisotropic versions of the Brezis-Van Schaftingen-Yung approach at $s=1$ and $s=0$

“…Similarly, the first inequality in (1.12) can be deduced from (1.13). The approach of Brezis-Van Schaftingen-Yung also inspires the work [8,37]. In this paper, we will give a positive answer to this question by the following theorem, which establishes anisotropic versions of formulas (1.10) and (1.12) and their limiting behaviors (1.11) and (1.13).…”