\(L^p\) gradient estimates and Calderón-Zygmund inequalities under Ricci lower bounds. (English) Zbl 1540.58013
Summary: 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 \(1 < p < +\infty \) 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 \(1 < p \leq 2\), analogous \(L^p\) bounds on higher even order Riesz transforms are obtained provided that also the derivatives of Ricci are controlled up to a suitable order.
MSC:
| 58J05 | Elliptic equations on manifolds, general theory |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 35B45 | A priori estimates in context of PDEs |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
Keywords:
\(L^p\) gradient estimates; Calderón-Zygmund inequalities; Riesz transform; integral Ricci bounds; harmonic coordinatesReferences:
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