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Riesz meets Sobolev

Thierry Coulhon, Adam Sikora (2010)

Colloquium Mathematicae

We show that the L p boundedness, p > 2, of the Riesz transform on a complete non-compact Riemannian manifold with upper and lower Gaussian heat kernel estimates is equivalent to a certain form of Sobolev inequality. We also characterize in such terms the heat kernel gradient upper estimate on manifolds with polynomial growth.

Riesz potentials and amalgams

Michael Cowling, Stefano Meda, Roberta Pasquale (1999)

Annales de l'institut Fourier

Let ( M , d ) be a metric space, equipped with a Borel measure μ satisfying suitable compatibility conditions. An amalgam A p q ( M ) is a space which looks locally like L p ( M ) but globally like L q ( M ) . We consider the case where the measure μ ( B ( x , ρ ) of the ball B ( x , ρ ) with centre x and radius ρ behaves like a polynomial in ρ , and consider the mapping properties between amalgams of kernel operators where the kernel ker K ( x , y ) behaves like d ( x , y ) - a when d ( x , y ) 1 and like d ( x , y ) - b when d ( x , y ) 1 . As an application, we describe Hardy–Littlewood–Sobolev type regularity theorems...

Riesz transform on manifolds and Poincaré inequalitie

Pascal Auscher, Thierry Coulhon (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We study the validity of the L p inequality for the Riesz transform when p > 2 and of its reverse inequality when 1 < p < 2 on complete riemannian manifolds under the doubling property and some Poincaré inequalities.

Riesz transforms on connected sums

Gilles Carron (2007)

Annales de l’institut Fourier

Assume that M 0 is a complete Riemannian manifold with Ricci curvature bounded from below and that M 0 satisfies a Sobolev inequality of dimension ν > 3 . Let M be a complete Riemannian manifold isometric at infinity to M 0 and let p ( ν / ( ν - 1 ) , ν ) . The boundedness of the Riesz transform of L p ( M 0 ) then implies the boundedness of the Riesz transform of L p ( M )

Some generic properties of nonlinear second order diffusional type problem

Vladimír Ďurikovič, Mária Ďurikovičová (1999)

Archivum Mathematicum

We are interested of the Newton type mixed problem for the general second order semilinear evolution equation. Applying Nikolskij’s decomposition theorem and general Fredholm operator theory results, the present paper yields sufficient conditions for generic properties, surjectivity and bifurcation sets of the given problem.

Some Gradient Estimates on Covering Manifolds

Nick Dungey (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

Let M be a complete Riemannian manifold which is a Galois covering, that is, M is periodic under the action of a discrete group G of isometries. Assuming that G has polynomial volume growth, we provide a new proof of Gaussian upper bounds for the gradient of the heat kernel of the Laplace operator on M. Our method also yields a control on the gradient in case G does not have polynomial growth.

Spectral shift and multiplicity of the first eigenvalue of the magnetic Schrödinger operator in two dimensions

László Erdős (2002)

Annales de l’institut Fourier

We show that the lowest eigenvalue of the magnetic Schrödinger operator on a line bundle over a compact Riemann surface M is bounded by the L 1 -norm of the magnetic field B . This implies a similar bound on the multiplicity of the ground state. An example shows that this degeneracy can indeed be comparable with M | B | even in case of the trivial bundle.

Stability results for Harnack inequalities

Alexander Grigor'yan, Laurent Saloff-Coste (2005)

Annales de l’institut Fourier

We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack inequalities under certain non-uniform changes of the weight. We also prove necessary and sufficient conditions for the Harnack inequalities to hold on complete non-compact manifolds having non-negative Ricci curvature outside a compact set and a finite first Betti number or just having asymptotically...

Submersions and equivariant Quillen metrics

Xiaonan Ma (2000)

Annales de l'institut Fourier

In this paper, we calculate the behaviour of the equivariant Quillen metric by submersions. We thus extend a formula of Berthomieu-Bismut to the equivariant case.

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