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

Abstract. With ~ ∆j ≥ 0 is the uniquely determined self-adjoint realization of the Laplace operator acting on j-forms on a geodesically complete Riemannian manifoldM 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 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 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 Vj = ~ ∆j − ∇†∇ of M is bounded from below (where V1 = Ric), which is shown to fail for large times if Vj is bounded. Based on these results, we formulate a conjecture on the boundedness of the covariant local Riesz-transform ∇(~ ∆j + κ) in L for all 1 ≤ p < ∞ (which we prove for 1 ≤ p ≤ 2), and explain its implications to geometric analysis, such as the L-Calderón-Zygmund inequality. Our main technical tool is a Bismut derivative formula for ∇e−t ∆j .