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

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 $$p > 2$$.