Truncations for fractional Laplacians

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a bounded Lipschitz domain. We prove and widely generalize a conjecture of A.\,I.~Nazarov \cite{Naz21}: for $s\in (1,\frac{3}{2})$ the quadratic form $Q_s^{\mathrm{SP}}[u]$ of the spectral fractional Dirichlet Laplacian strictly increases under the map $u\mapsto |u|$ provided $u\in {\tilde{H}}^s(\Omega )$ changes sign in $\Omega$.

7 pages