\(H^1 - L^1\) boundedness of Riesz transforms on Riemannian manifolds and on graphs
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 \textit{L. Saloff-Coste} [Potential Anal. 4, No. 4, 429-467 (1995; Zbl 0840.31006)] in the case of manifolds, and by \textit{T. Delmotte} [Rev. Mat. Iberoam. 15, No. 1, 181-232 (1999; Zbl 0922.60060)] in the case of graphs.
scientific article; zbMATH DE number 1618881