Riesz transform and related inequalities on non‐compact Riemannian manifolds

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Riesz transform on manifols and Poincaré inequalities (2005, background): “It is proved in [10] that for any q ∈ (1, 2), there exists a complete Riemannian manifold with (D) such that (R Rp) fails for all 1 < p < q.”
The Calder\'on-Zygmund inequality and Sobolev spaces on noncompact Riemannian manifolds (2014, background): “1 of [9] we have the multiplicative inequality ‖grad(u)‖p ≤ C(p) ‖u‖ 1 2 p ‖∆u‖ 1 2 p , (11) which completes the proof in this case, using once more inequality (10).”
Riesz transform on manifolds and Poincaré inequalitie (2005, methodology): “This is exactly the obstacle that we get around in Section 1 with a modified Calder´ on-Zygmund decomposition and it is not clear that the same ideas can be employed under the assumption taken in [ 10 ].”
Gaussian heat kernel upper bounds via the Phragmén–Lindelöf theorem (2006, unknown): “t > 0, x ∈ M, 2It is worth noting at this point that there is a connection between estimates of the heat kernel on 1-forms and the Lp boundedness of the Riesz transform for p>2 (see for example [21, 20, 51]). 4 THIERRY COULHON AND ADAM SIKORA for some Kand D > 0. Then (1.4) |pz(x,y)| ≤ eK(R”
Riesz transform on manifolds and heat kernel regularity (2004, methodology): “Another direction is to pursue [ 20 ] by using Littlewood-Paley-Stein functionals and prove the conjecture stated there.”