Weighted norm estimates and $L_{p}$-spectral independence of linear operators

Peer C. Kunstmann¹ ; Hendrik Vogt²

¹ Mathematisches Institut I Universität Karlsruhe Englerstraße 2 D-76128 Karlsruhe, Germany
² Institut für Analysis Fachrichtung Mathematik Technische Universität Dresden D-01062 Dresden, Germany

Colloquium Mathematicum, Tome 109 (2007) no. 1, pp. 129-146.

Résumé

We investigate the $L_p$-spectrum of linear operators defined consistently on $L_p({\mit\Omega} )$ for $p_0\le p\le p_1$, where $({\mit\Omega} ,\mu )$ is an arbitrary $\sigma $-finite measure space and $1\le p_0 p_1\le \infty $. We prove $p$-independence of the $L_p$-spectrum assuming weighted norm estimates. The assumptions are formulated in terms of a measurable semi-metric $d$ on $({\mit\Omega} ,\mu )$; the balls with respect to this semi-metric are required to satisfy a subexponential volume growth condition. We show how previous results on $L_p$-spectral independence can be treated as special cases of our results and give examples—including strictly elliptic operators in Euclidean space and generators of semigroups that satisfy (generalized) Gaussian bounds—to indicate improvements.

Détail

DOI : 10.4064/cm109-1-11

Keywords: investigate p spectrum linear operators defined consistently mit omega where mit omega arbitrary sigma finite measure space infty prove p independence p spectrum assuming weighted norm estimates assumptions formulated terms measurable semi metric mit omega balls respect semi metric required satisfy subexponential volume growth condition previous results p spectral independence treated special cases results examples including strictly elliptic operators euclidean space generators semigroups satisfy generalized gaussian bounds indicate improvements