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

  • Jocelyn Magniez
  • El Maati OuhabazEmail author
Article
  • 13 Downloads

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 \(p > 2\).

Keywords

Hodge–de Rham Laplacian Differential forms Schrodinger operators Heat kernels Riesz transform 

Mathematics Subject Classification

58J35 42B20 47D03 

Notes

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© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Institut de Mathématiques de Bordeaux, UMR 5251Université de BordeauxTalenceFrance

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