L p -estimates for the heat semigroup on differential forms, and related problems

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 -- $\rightarrow$ $\Delta$ k be the Hodge-de Rham Laplacian on differential k-forms with k $\ge$ 1. By the Bochner decomposition formula -- $\rightarrow$ $\Delta$ k = * + R k. Under the assumption that the negative part R -- k is in an enlarged Kato class, we prove that for all p $\in$ [1, $\infty$], e --t -- $\rightarrow$ $\Delta$ k p--p $\le$ 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 -- $\rightarrow$ $\Delta$ k) t>0 is not uniformly bounded on L p for any p = 2. We also prove the gradient estimate e --t$\Delta$ p--p $\le$ Ct -- 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.