Sharp Boundedness Criteria for the Wolff Potential on Complete Riemannian Manifolds.

Source abstract

Let \mathcal{M} be a complete Riemannian manifold with \mathrm{Ric}\geq 0. We characterize the strong-type (\mathfrak{p},\mathfrak{q}) boundedness of the Wolff potential \mathcal{W}_{\upalpha,\mathfrak{p}} by Euclidean volume growth, we prove that \nabla\mathcal{W}_{\upalpha,\mathfrak{p}} is of weak type (1,1) in analogy with Calderon-Zygmund operators, and we establish a no-go theorem showing that strictly sub-Euclidean volume growth \mathcal{V}_{\mathfrak{u}}(r)\leq Cr^{\upbeta} with \upbeta<\mathfrak{n} completely obstructs any L^{\mathfrak{p}}-L^{\mathfrak{q}} bound, regardless of the choice of exponents.