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L 1 -estimates for eigenfunctions and heat kernel estimates for semigroups dominated by the free heat semigroup

L 1 -estimates for eigenfunctions and heat kernel estimates for semigroups dominated by the free... We investigate selfadjoint positivity preserving C 0 -semigroups that are dominated by the free heat semigroup on $${\mathbb{R}^d}$$ R d . Major examples are semigroups generated by Dirichlet Laplacians on open subsets or by Schrödinger operators with absorption potentials. We show explicit global Gaussian upper bounds for the kernel that correctly reflect the exponential decay of the semigroup. For eigenfunctions of the generator that correspond to eigenvalues below the essential spectrum, we prove estimates of their L 1 -norm in terms of the L 2 -norm and the eigenvalue counting function. This estimate is applied to a comparison of the heat content with the heat trace of the semigroup. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

L 1 -estimates for eigenfunctions and heat kernel estimates for semigroups dominated by the free heat semigroup

Journal of Evolution Equations , Volume 15 (4) – Dec 1, 2015

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References (10)

Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer Basel
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-015-0285-3
Publisher site
See Article on Publisher Site

Abstract

We investigate selfadjoint positivity preserving C 0 -semigroups that are dominated by the free heat semigroup on $${\mathbb{R}^d}$$ R d . Major examples are semigroups generated by Dirichlet Laplacians on open subsets or by Schrödinger operators with absorption potentials. We show explicit global Gaussian upper bounds for the kernel that correctly reflect the exponential decay of the semigroup. For eigenfunctions of the generator that correspond to eigenvalues below the essential spectrum, we prove estimates of their L 1 -norm in terms of the L 2 -norm and the eigenvalue counting function. This estimate is applied to a comparison of the heat content with the heat trace of the semigroup.

Journal

Journal of Evolution EquationsSpringer Journals

Published: Dec 1, 2015

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