Kernel estimates and Lp-spectral independence of generators of C0-semigroups

After the appearance ofW. Arendt’s result that “Gaussian estimate of a semigroup implies the Lp-spectral independence of the generator,” various generalizations have been obtained. This paper shows that a certain kernel estimate of a semigroup implies the Lp-spectral independence of the generator, generalizing the case of upper Gaussian estimate and “Gaussian estimate of order α ∈ (0, 1] [S. Miyajima, H. Shindoh, Gaussian estimates of order α and Lp-spectral independence of generators of C0-semigroups, Positivity 11 (1) (2007) 15–39], Definition 3.1.” The proof uses S. Karrmann’s result about the Lp-spectral independence and B.A. Barnes’ theorem about the spectrum of integral operators. As an application, the Lp-spectral independence of −[(−Δ)α + V ] (α ∈ (0, 1]) for a suitable V is proved with the help of a recent result by V. Liskevich, H. Vogt and J. Voigt [V. Liskevich, H. Vogt, J. Voigt, Gaussian bounds for propagators perturbed by potentials, J. Funct. Anal. 238 (2006) 245–277].