On the weak \(L ^{p }\)-Hodge decomposition and Beurling--Ahlfors transforms on complete Riemannian manifolds (Q718878)

The author proves a weak \(L^p\)-Hodge decomposition on a (stochastically) complete Riemannian manifold \(M\) with non-negative Weitzenböck curvature on \(k\)-forms, based on the \(L^p\) boundedness of the Beurling-Ahlfors transform. In particular, the boundedness of the Riesz transforms is not needed. As a consequence, a logarithmic Sobolev inequality is obtained in dimension \(n\geq 3\) for vanishing Ricci curvature. The proofs are based on a martingale representation formula for \(dd^* ( a + \square )^{-1}\), where \(a>0\) and \(\square\) denotes the Hodge Laplacian.