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L^p-Estimates for the Heat Semigroup on Differential Forms, and Related Problems

10$
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L^p-Estimates for the Heat Semigroup on Differential Forms, and Related Problems

10$

Jocelyn Magniez & El Maati Ouhabaz 

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 \overrightarrow{\Delta }_k be the Hodge–de Rham Laplacian on differential k-forms with k \ge 1. By the Bochner decomposition formula, \overrightarrow{\Delta }_k = \nabla ^* \nabla + R_k, where \nabla denotes the Levi-Civita connection and R_k is a symmetric section of \mathrm{End}(\Lambda ^kT^*M). Under the assumption that the negative part R_k^- is in an enlarged Kato class, we prove that for all p \in [1, \infty ], \Vert e^{-t\overrightarrow{\Delta }_k}\Vert _{p-p} \le C ( t \log t)^{\frac{D}{4}(1- \frac{2}{p})} (for large t), where D is a homogeneous “dimension” appearing in the volume doubling property. This estimate can be improved if R_k^- is strongly sub-critical. In general, (e^{-t\overrightarrow{\Delta }_k})_{t>0} is not uniformly bounded on L^p for any p \not = 2. We also prove the gradient estimate \Vert \nabla e^{-t\Delta }\Vert _{p-p} \le C t^{-\frac{1}{p}}, where \Delta 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.

Only units of this product remain
Year 2020
Language English
Format PDF
DOI 10.1007/s12220-019-00188-1