With $$\vec {\Delta }_j\ge 0$$ Δ → j ≥ 0 is the uniquely determined self-adjoint realization of the Laplace operator acting on j-forms on a geodesically complete Riemannian manifold M and $$\nabla $$ ∇ the Levi-Civita covariant derivative, we prove among other things a Gaussian heat kernel bound for $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$ ∇ e - t Δ → j , if the curvature tensor of M and its covariant derivative are bounded, an exponentially weighted $$L^p$$ L p -bound for the heat kernel of $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$ ∇ e - t Δ → j , if the curvature tensor of M and its covariant derivative are bounded, that $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$ ∇ e - t Δ → j is bounded in $$L^p$$ L p for all $$1\le p<\infty $$ 1 ≤ p < ∞ , if the curvature tensor of M and its covariant derivative are bounded, a second order Davies-Gaffney estimate (in terms of $$\nabla $$ ∇ and $$\vec {\Delta }_j$$ Δ → j ) for $$\mathrm {e}^{ -t\vec {\Delta }_j }$$ e - t Δ → j for small times, if the j-th degree Bochner-Lichnerowicz potential $$V_j=\vec {\Delta }_j-\nabla ^{\dagger }\nabla $$ V j = Δ → j - ∇ † ∇ of M is bounded from below (where $$V_1=\mathrm {Ric}$$ V 1 = Ric ), which is shown to fail for large time, if $$V_j$$ V j is bounded. Based on these results, we formulate a conjecture on the boundedness of the covariant local Riesz-transform $$\nabla (\vec {\Delta }_j+\kappa )^{-1/2}$$ ∇ ( Δ → j + κ ) - 1 / 2 in $$L^p$$ L p for all $$1\le p<\infty $$ 1 ≤ p < ∞ (which we prove for $$1\le p\le 2$$ 1 ≤ p ≤ 2 ), and explain its implications to geometric analysis, such as the $$L^p$$ L p -Calderón-Zygmund inequality. Our main technical tool is a Bismut derivative formula for $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$ ∇ e - t Δ → j .

We prove using an integral criterion the existence and completeness of the wave operators 𝑊 ± (Δ𝑔,ℎ ) corresponding to the Hodge-Laplacians Δ (𝑝) 𝜈 acting on differential 𝑝-forms, for 𝜈 ∈ {𝑔, ℎ}, induced by two quasi-isometric Riemannian metrics 𝑔 and ℎ on an complete open smooth manifold 𝑀. In particular, this result provides a criterion for the absolutely continuous spectra 𝜎 ac (Δ𝜈 to coincide. The proof is based on gradient estimates obtained by probabilistic Bismut-type formulae for the heat semigroup defined by spectral calculus. By these localised formulae, the integral criterion only requires local curvature bounds and some upper local control on the heat kernel acting on functions, but no control on the injectivity radii. A consequence is a stability result of the absolutely continuous spectrum under a Ricci flow. As an application we concentrate on the important case of conformal perturbations.

With ∆ j ≥ 0 is the uniquely determined self-adjoint realization of the Laplace operator acting on j-forms on a geodesically complete Riemannian manifold M and ∇ the Levi-Civita covariant derivative, we prove amongst other things• a Li-Yau type heat kernel bound for ∇e −t ∆j , if the curvature tensor of M and its covariant derivative are bounded, • an exponentially weighted L p bound for the heat kernel of ∇e −t ∆j , if the curvature tensor of M and its covariant derivative are bounded, • that ∇e −t ∆j is bounded in L p for all 1 ≤ p < ∞, if the curvature tensor of M and its covariant derivative are bounded, • a second order Davies-Gaffney estimate (in terms of ∇ and ∆ j ) for e −t ∆j for small times, if the j-th degree Bochner-Lichnerowicz potential V j = ∆ j − ∇ † ∇ of M is bounded from below (where V 1 = Ric), which is shown to fail for large times if V j is bounded. Based on these results, we formulate a conjecture on the boundedness of the covariant local Riesz-transform ∇( ∆ j + κ) −1/2 in L p for all 1 ≤ p < ∞ (which we prove for 1 ≤ p ≤ 2), and explain its implications to geometric analysis, such as the L p -Calderón-Zygmund inequality. Our main technical tool is a Bismut derivative formula for ∇e −t ∆j .

No abstract