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Riesz transform and related inequalities on non-compact Riemannian manifolds. (English) Zbl 1037.58017
The authors of this interesting paper consider the question when the inequality \[ \| | \nabla f| \| _p\leq C_p\| \Delta^{1/2}f\| _p\quad f\in C^\infty_0(M) \] holds on a complete Riemannian manifold \(M\). The inequality is the harder part of the proof of the equivalence of two classical norms in a first order Sobolev space. They conjecture that the inequality holds if \(1<p\leq 2\) or \(2<p<\infty\) and a heat semi-group on \(M\) satisfies an domination condition. The weaker version with \(\sqrt{\| f\| _p\| \Delta f\| _p}\) instead of \(\| \Delta^{1/2}f\| _p\) is proved. A generalization with different integral exponent on the left and right sides is given. The original inequality is proved under stronger assumptions. Namely it is proved that if \(M\) has doubling property, an upper estimate for the heat kernel holds as well as some estimate for the heat kernel on 1-forms then the Riesz transform \(\nabla\Delta^{-1/2}\) is bounded in \(L^p\) for \(2<p<\infty\). This implies the original inequality.

MSC:
58J35 Heat and other parabolic equation methods for PDEs on manifolds
44A15 Special integral transforms (Legendre, Hilbert, etc.)
47B34 Kernel operators
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