L p -estimates for the heat semigroup on differential forms, and related problems
Language
en
Article de revue
This item was published in
Journal of Geometric Analysis. 2020, vol. 30, n° 3, p. 3002-3025
English Abstract
We consider a complete non-compact Riemannian manifold satisfying the volume doubling property and a Gaussian upper bound for its heat kernel (on functions). Let − → ∆ k be the Hodge-de Rham Laplacian on differential k-forms ...Read more >
We consider a complete non-compact Riemannian manifold satisfying the volume doubling property and a Gaussian upper bound for its heat kernel (on functions). Let − → ∆ k be the Hodge-de Rham Laplacian on differential k-forms with k ≥ 1. By the Bochner decomposition formula − → ∆ k = * + R k. Under the assumption that the negative part R − k is in an enlarged Kato class, we prove that for all p ∈ [1, ∞], e −t − → ∆ k p−p ≤ C(t log t) D 4 (1− 2 p) (for large t). This estimate can be improved if R − k is strongly sub-critical. In general, (e −t − → ∆ k) t>0 is not uniformly bounded on L p for any p = 2. We also prove the gradient estimate e −t∆ p−p ≤ Ct − 1 p , where ∆ is the Laplace-Beltrami operator (acting on functions). Finally we discuss heat kernel bounds on forms and the Riesz transform on L p for p > 2.Read less <
Origin
Hal imported