Estimates for functions of the Laplacian on manifolds with bounded geometry

In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below and positive injectivity radius. Denote by L the Laplace-Beltrami operator on M. We assume that the kernel associated to the heat semigroup generated by L satisfies a mild decay condition at infinity. We prove that if m is a bounded holomorphic function in a suitable strip of the complex plane, and satisfies Mihlin-Hormander type conditions of appropriate order at infinity, then the operator m(L) extends to an operator of weak type 1. This partially extends a celebrated result of J. Cheeger, M. Gromov and M. Taylor, who proved similar results under much stronger curvature assumptions on M, but without any assumption on the decay of the heat kernel.Comment: 19 page

Higher order Riesz transforms on noncompact symmetric spaces

In this note we prove various sharp boundedness results on suitable Hardy type spaces for Riesz transforms of arbitrary order on noncompact symmetric spaces of arbitrary rank.Comment: v2: the first version has been revised and splitted up in two papers, of which this new version is one par

Revealing interference by continuous variable discordant states

In general, a pair of uncorrelated Gaussian states mixed in a beam splitter produces a correlated state at the output. However, when the inputs are identical Gaussian states the output state is equal to the input, and no correlations appear, as the interference had not taken place. On the other hand, since physical phenomena do have observable effects, and the beam splitter is there, a question arises on how to reveal the interference between the two beams. We prove theoretically and demonstrate experimentally that this is possible if at least one of the two beams is prepared in a discordant, i.e. Gaussian correlated, state with a third beam. We also apply the same technique to reveal the erasure of polarization information. Our experiments involves thermal states and the results show that Gaussian discordant states, even when they show a positive Glauber P-function, may be useful to achieve specific tasks.Comment: published versio

Toward third order ghost imaging with thermal light

Recently it has been suggested that an enhancement in the visibility of ghost images obtained with thermal light can be achieved exploiting higher order correlations [3]. This paper reports on the status of an higher order ghost imaging experiment carried on at INRIM labs exploiting a pseudo-thermal source and a CCD camera.Comment: To be published in Proceedings of Recent advances in Foundations of Quantum Mechanics and Quantum Informatio