Maximum principles and spectral analysis for the superposition of operators of fractional order

Abstract

We consider a "superposition operator" obtained through the continuous superposition of operators of mixed fractional order, modulated by a signed Borel finite measure defined over the set $[0,1]$. The relevance of this operator is rooted in the fact that it incorporates special and significant cases of interest, like the mixed operator $-\hat{I}”+(-\hat{I}”{)}^s$, the (possibly) infinite sum of fractional Laplacians and allows to consider operators carrying a "wrong sign". We first outline weak and strong maximum principles for this type of operators. Then, we complete the spectral analysis for the related Dirichlet eigenvalue problem started in [DPLSV25b].