L^p gradient estimates and Calderón--Zygmund inequalities under Ricci lower bounds

2022/07/18 by Marini, Ludovico, Meda, Stefano, Pigola, Stefano +1
#35B45 #42B20 #53C21 #58J05 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2207.08545

Abstract

In this paper we investigate the validity of first and second order L^p estimates for the solutions of the Poisson equation depending on the geometry of the underlying manifold. We first present L^p 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 L^p estimates for the second order Riesz transform (or, equivalently, the validity of L^p Calderón--Zygmund inequalities) on the whole scale 1

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