Duke Mathematical Journal

Riesz transform and $L^p$-cohomology for manifolds with Euclidean ends

Gilles Carron, Thierry Coulhon, and Rew Hassell

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Abstract

Let $M$ be a smooth Riemannian manifold that is the union of a compact part and a finite number of Euclidean ends, $\mathbb{R}^n {\setminus} B(0,R)$ for some $R > 0$, each of which carries the standard metric. Our main result is that the Riesz transform on $M$ is bounded from $L^p(M) \to L^p(M; T^*M)$ for $1\lt p \lt n$ and unbounded for $p \geq n$ if there is more than one end. It follows from known results that in such a case, the Riesz transform on $M$ is bounded for $1\lt p\leq 2$ and unbounded for $p \gt n$; the result is new for $2\lt p\leq n$. We also give some heat kernel estimates on such manifolds.

We then consider the implications of boundedness of the Riesz transform in $L^p$ for some $p > 2$ for a more general class of manifolds. Assume that $M$ is an $n$-dimensional complete manifold satisfying the Nash inequality and with an $O(r^n)$ upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on $L^p$ for some $p > 2$ implies a Hodge–de Rham interpretation of the $L^p$-cohomology in degree $1$ and that the map from $L^2$- to $L^p$-cohomology in this degree is injective

Article information

Source
Duke Math. J. Volume 133, Number 1 (2006), 59-93.

Dates
First available in Project Euclid: 19 April 2006

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1145452056

Digital Object Identifier
doi:10.1215/S0012-7094-06-13313-6

Mathematical Reviews number (MathSciNet)
MR2219270

Zentralblatt MATH identifier
1106.58021

Subjects
Primary: 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]
Secondary: 58J35: Heat and other parabolic equation methods 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)

Citation

Carron, Gilles; Coulhon, Thierry; Hassell, Rew. Riesz transform and L p -cohomology for manifolds with Euclidean ends. Duke Math. J. 133 (2006), no. 1, 59--93. doi:10.1215/S0012-7094-06-13313-6. https://projecteuclid.org/euclid.dmj/1145452056.


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