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

Tom 109 / 2007

Peer C. Kunstmann, Hendrik Vogt Colloquium Mathematicum 109 (2007), 129-146 MSC: 35P05, 47A10, 47D03. DOI: 10.4064/cm109-1-11


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.


  • Peer C. KunstmannMathematisches Institut I
    Universität Karlsruhe
    Englerstraße 2
    D-76128 Karlsruhe, Germany
  • Hendrik VogtInstitut für Analysis
    Fachrichtung Mathematik
    Technische Universität Dresden
    D-01062 Dresden, Germany

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