Real Analyticity of Solutions of Hamilton's Equation.
We show that the 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.
Let be a metric space, equipped with a Borel measure satisfying suitable compatibility conditions. An amalgam is a space which looks locally like but globally like . We consider the case where the measure of the ball with centre and radius behaves like a polynomial in , and consider the mapping properties between amalgams of kernel operators where the kernel behaves like when and like when . As an application, we describe Hardy–Littlewood–Sobolev type regularity theorems...
We study the validity of the inequality for the Riesz transform when and of its reverse inequality when on complete riemannian manifolds under the doubling property and some Poincaré inequalities.
Assume that is a complete Riemannian manifold with Ricci curvature bounded from below and that satisfies a Sobolev inequality of dimension . Let be a complete Riemannian manifold isometric at infinity to and let . The boundedness of the Riesz transform of then implies the boundedness of the Riesz transform of
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.
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.
We show that the lowest eigenvalue of the magnetic Schrödinger operator on a line bundle over a compact Riemann surface is bounded by the -norm of the magnetic field . This implies a similar bound on the multiplicity of the ground state. An example shows that this degeneracy can indeed be comparable with even in case of the trivial bundle.
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...
We prove Strichartz inequalities for the solution of the Schrödinger equation related to the full Laplacian on the Heisenberg group. A key point consists in estimating the decay in time of the norm of the free solution; this requires a careful analysis due also to the non-homogeneous nature of the full Laplacian.
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.