Abstract
We consider a family of semiclassically scaled second-order elliptic differential operators on high tensor powers of a Hermitian line bundle (possibly, twisted by an auxiliary Hermitian vector bundle of arbitrary rank) on a Riemannian manifold of bounded geometry. We establish an off-diagonal Gaussian upper bound for the associated heat kernel. The proof is based on some tools from the theory of operator semigroups in a Hilbert space, results on Sobolev spaces adapted to the current setting, and weighted estimates with appropriate exponential weights.
Highlights
We consider a family of second-order elliptic differential operators D p, p ∈ N, acting on sections of tensor powers L p of a Hermitian line bundle L of bounded geometry on a complete Riemannian manifold of bounded geometry
One can take an auxiliary Hermitian vector bundle E and consider operators acting on sections of
Similar operators have been studied in connection with the Demailly holomorphic Morse inequalities for the Dolbeault cohomology associated with high tensor powers of a holomorphic Hermitian line bundle over a compact complex manifold [2]
Summary
We consider a family of second-order elliptic differential operators D p , p ∈ N, acting on sections of tensor powers L p of a Hermitian line bundle L of bounded geometry on a complete Riemannian manifold of bounded geometry. Similar operators have been studied in connection with the Demailly holomorphic Morse inequalities for the Dolbeault cohomology associated with high tensor powers of a holomorphic Hermitian line bundle over a compact complex manifold [2] (see [3,4,5] and references therein) They may be viewed as elliptic models for the geometric Fokker–Planck operators (see, for instance, [6,7]). The main result of the paper is an off-diagonal Gaussian upper bound for the heat kernel associated with the operator D p in the semiclassical limit p → ∞ Such Gaussian estimates are, well known for a fixed p (see, for instance, [8] for results on manifolds of bounded geometry).
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