Riesz transform on manifolds and heat kernel regularity

By P. Auscher, T. Coulhon, Xuan Thinh Duong, S. Hofmann. 2004

317 citations recorded by the source corpus.

Abstract One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is L p bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain L p estimate in the same interval of p's.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b75721d254c88fe6f67e097eb31bd4c6d54cd7a1

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Citations in the corpus

Sharp weighted norm estimates beyond Calderón–Zygmund theory (2015, unknown): “We point out that (b) and (c) are the main assumptions and were already used in numerous works to replace the notion of Calderón–Zygmund operators (see, e.g., [Auscher 2007; Auscher et al. 2004] and references therein).”
The Incompressible Navier–Stokes Equations on Non-compact Manifolds (2014, background): “These kind of estimates are related to the behaviour of the heat kernel which is well studied in the literature for various types of manifolds for both the Laplace-Beltrami and the Hodge Laplacian (see for example [38], [2], [3], [25], [41], [47], [56], [10], [12], [4], [48], [6] and others) The mai”
Heat Kernel Bounds on Metric Measure Spaces and Some Applications (2014, methodology): “y using heat kernel gradient estimates Now we begin the proof of the main results by combing the heat kernel gradient estimate in Corollaries 1.1 and 1.2, and applying the original method of [14] and [9]. Proof of Theorem 5.1. For each f ∈ V(X), it follows from the deﬁnition of V(X) that, k∇(−∆)−1/2”
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.”
Gradient estimates for heat kernels and harmonic functions (2017, background): “Further, it was discovered in [7] that the weaker L-version of these estimates governs the boundedness of Riesz transforms on L in an interval (2, p0), for 2 < p0 < +∞.”
New Abstract Hardy Spaces (2007, background): “For example H(1) 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].”
Hypoelliptic heat kernel inequalities on the Heisenberg group (2005, background): “and are measurable functions defined almost everywhere on [0, 1 ] k ×W . We will denote the domain of D k in L p � [0, 1] k × W � by D k,p , which is the completion of the family”
Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights (2006, methodology): “The third step was taken in [7], inspired by the good-λ estimates in the Ph.”
A characterization of Hajłasz–Sobolev and Triebel–Lizorkin spaces via grand Littlewood–Paley functions☆ (2009, background): “2 apply in a wide range of settings, for instance, to Ahlfors n-regular metric measure spaces (see [22]), d-spaces (see [35]), Lie groups of the polynomial volume growth (see [36, 37, 29, 2]), the complete connected noncompact manifolds with a doubling measure (see [4, 5]), compact Carnot-Carathéodo”
Gaussian heat kernel upper bounds via the Phragmén–Lindelöf theorem (2006, unknown): “In particular, it is proved in [4] that under the assumptions below, for a = 1/2, the Riesz transform is bounded on L(M, dμ) for 2 < p < +∞.”
Some gradient estimates for the heat equation on domains and for an equation by Perelman (2006, methodology): “ for the heat equation in both the ﬁxed metric and the Ricci ﬂow case. Gradient estimates for the heat equation are important tools in geometric analysis as pointed out in the papers [LY], [H], [CH], [ACDH] and others. In this paper, for the ﬁxed metric case, two gradient estimates are proven. One i”
Heat semigroup and singular PDEs (2015, background): “y recalled in section B.1.3. B.1.1 Lq-estimates of the gradient of the semigroup Given q0 > 2, the uniform Lq0-boundedness of the gradient (or “carr´e du champ”) of the semigroup was introduced in [2] (Gp0) sup t>0 k √ tΓe−tLk q0→q0 <+∞. By deﬁnition of the carr´e du champ operator, (G2) ho”