gradient estimates and Calderón–Zygmund inequalities under Ricci lower bounds
Ludovico Marini
Università degli Studi di Milano-Bicocca, Milano, ItalyStefano Meda
Università degli Studi di Milano-Bicocca, Milano, ItalyStefano Pigola
Università degli Studi di Milano-Bicocca, Milano, ItalyGiona Veronelli
Università degli Studi di Milano-Bicocca, Milano, Italy
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Abstract
In this paper, we investigate the validity of first and second order estimates for the solutions of the Poisson equation depending on the geometry of the underlying manifold. We first present estimates of the gradient under the assumption that the Ricci tensor is lower bounded in a local integral sense, and construct the first counterexample showing that they are false, in general, without curvature restrictions. Next, we obtain estimates for the second order Riesz transform (or, equivalently, the validity of Calderón–Zygmund inequalities) on the whole scale by assuming that the injectivity radius is positive and that the Ricci tensor is either pointwise lower bounded, or non-negative in a global integral sense. When , analogous bounds on higher even order Riesz transforms are obtained provided that also the derivatives of Ricci are controlled up to a suitable order.
Cite this article
Ludovico Marini, Stefano Meda, Stefano Pigola, Giona Veronelli, gradient estimates and Calderón–Zygmund inequalities under Ricci lower bounds. Rev. Mat. Iberoam. 40 (2024), no. 3, pp. 803–826
DOI 10.4171/RMI/1476