Conjecture on Gaussian estimates and Riesz transform boundedness under Kato curvature conditions

Let $M$ be a non-parabolic Riemannian manifold, endowed with its Riemannian measure, satisfying the volume-doubling condition and an upper Gaussian estimate, denoted by

$$and$$

, respectively. Let $\vec{\Delta}$ be the Hodge Laplacian on differential forms, and suppose it is strongly positive. Assume that the negative part of the Ricci curvature, denoted by $|\mathrm{Ric}_-|$, satisfies the Kato condition $(K^\infty,1)$.

Gaussian estimates and Riesz transform conjecture. The heat kernel of the Hodge Laplacian should have Gaussian estimates, and the Riesz transform should be bounded on $L^p$ for every $p\in (1,\infty)$.

The conjecture seeks to extend the preceding theorem, which establishes the same conclusions under separate hypotheses on the lowest Ricci eigenvalue and the Schrödinger operator. Its validity would give Gaussian heat-kernel control and $L^p$ boundedness of the Riesz transform from strong positivity of the Hodge Laplacian together with a Kato condition on the negative Ricci curvature.