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Riesz transform and \(L^p\)-cohomology for manifolds with Euclidean ends. (English) Zbl 1106.58021
The authors are here interested in studying boundedness properties of the Riesz transform on some particular Riemannian manifolds. More precisely, they consider smooth Riemannian manifolds \(M\) that are 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. The Riesz transform on \(M\) is the operator \(T:L^2(M)\to L^2(M;T^*M)\), given by \(T(f)=d\triangle^{-1/2}f\), where \(\triangle\) is the positive Laplace operator on \(M\). It is just well known that \(T\) is always a bounded map for \(p=2\). Their main result is that \(T\) is bounded for \(1<p<n\) and unbounded for \(p\geq n\), if there is more than one end. The method used is to analyze the kernel of \(\triangle^{-1/2}\) to which can be applied the theory of scattering differential operators [Melrose]. Then, analyzing also the kernel of \(T\) they directly arrive to obtain the result.
A further result is contained in this paper that relates the boundedness of the Riesz transform in \(L^p\), \(p>2\), to a more general class of manifolds (\(n\)-dimensional complete manifolds satisfying the Nash inequality and under an \(O(r^n)\) upper bound on the volume growth of geodesic balls). They show, in fact, that under suitable hypotheses, the Riesz transform is bounded in \(L^p\) for some \(p>2\), and for the first space of reduced \(L^p\) cohomology of \((M,g)\), i.e., \(H^1_p(M)={{\{\alpha\in L^p(T^*M), d\alpha=0\}}\over{\overline{dC^\infty_0(M)}}}\), one has the isomorphism \(H^1_p(M)\cong\{\alpha\in L^p(M;T^*M)| d\alpha=0\), \(d^*\alpha=0\}\).
The paper ends with some concluding remarks and a list of open problems.

MSC:
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J35 Heat and other parabolic equation methods for PDEs on manifolds
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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