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\(H^1 - L^1\) boundedness of Riesz transforms on Riemannian manifolds and on graphs. (English) Zbl 0982.42008

The author proves that the Riesz transforms are bounded from \(H^1\) to \(L^1\) on complete manifolds and on graphs with the doubling property and the Poincaré inequality. Since the Riesz transforms are bounded on \(L^2\) and are given by a kernel \(K(x,y)\), it suffices to show that \(K(x,y)\) satisfies a Hörmander integral condition. To prove this result, the author uses the Hörmander regularity for solutions of the heat equations, which is a consequence of the parabolic Harnack principle, shown by L. Saloff-Coste [Potential Anal. 4, No. 4, 429-467 (1995; Zbl 0840.31006)] in the case of manifolds, and by T. Delmotte [Rev. Mat. Iberoam. 15, No. 1, 181-232 (1999; Zbl 0922.60060)] in the case of graphs.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B30 \(H^p\)-spaces
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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