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Riesz transform on manifolds and heat kernel regularity. (English) Zbl 1086.58013
The purpose of the paper under review is to give a necessary and sufficient condition for the two natural definitions of homogeneous \(L^p\) Sobolev spaces to coincide on a large class of Riemannian manifolds, for \(p\in (q_0,p_0)\), where \(2<p_0\leq \infty\) and \(q_0\) is the conjugate exponent of \(p_0\).
The paper is structured into 6 sections.
In the first section, the authors recall previous results which will be used in the following and state the main results of this paper.
Let \(M\) be a complete non-compact Riemannian manifold. First, they prove that, under certain natural conditions, the Riesz transform is bounded on \(L^p\), for \(1<p<2\).
Using the local and global criterion for singular integrals, developed in section 2, the above result is extended for any \(1<p\leq \infty\) (see sections 3 and 4).
As applications of these results, \(L^p\) Hodge decomposition for non-compact manifolds is stated in section 5.

MSC:
58J35 Heat and other parabolic equation methods for PDEs on manifolds
58J10 Differential complexes
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35B65 Smoothness and regularity of solutions to PDEs
35K05 Heat equation
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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