H1–L1 Boundedness of Riesz Transforms on Riemannian Manifolds and on Graphs

By E. Russ. 2001

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Hardy Spaces of Differential Forms on Riemannian Manifolds (2006, result): “). Also, our Hardy spaces are designed so that the Riesz transform is automatically bounded on them. Specializing to speciﬁc situations allows us to recover results obtained by the third author alone [38] or with M. Marias [33]. We now describe precisely our setting. Let Mbe a complete Riemannian ma”
Gaussian estimates and Lp-boundedness of Riesz means (2002, background): “On the other hand, some more subtle phenomena or limit cases seem to require the stronger Poincar´ e inequalities (see [40], [ 41 ]).”
New Abstract Hardy Spaces (2007, background): “pace [1, 2, 8, 20, 17, 19, 21, 18, 22, 27]. For example H1 CW(X) is not well adapted to operators such as the Riesz transform on Riemannian manifolds (or on graphs) or the maximal regularity operator [1, 2, 4, 27, 29]. That is why in [17, 27] the authors deﬁne a new space H1 L by the L 1(X) norm of ”
H1 and BMO for certain locally doubling metric measures spaces (2009, background): “For instance, see [40] and the references therein for the equivalence between a scale-invariant parabolic Harnack inequality and a scaled Poincar´e inequality, and [2, 16, 39] for recents results on the boundedness of Riesz transforms on such manifolds.”