Estimates for the covariant derivative of the heat semigroup on differential forms, and covariant Riesz transforms

Abstract

AbstractWith $$\vec {\Delta }_j\ge 0$$ 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 .