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Reverse inequalities for quasi-Riesz transform on the Vicsek cable system
No full textThis work is devoted to the study of so-called ``reverse Riesz'' inequalities and suitable variants in the context of some fractal-like cable systems. It was already proved by L. Chen, T. Coulhon, J. Feneuil and the second author that, in the Vicsek cable system, the inequality is false for all . Following a recent joint paper by the two authors and M. Yang, we examine the validity of ``reverse quasi-Riesz'' inequalities, of the form , in the (unbounded) Vicsek cable system, for and . These reverse inequalities are strongly related to the problem of boundedness of the operators , the so-called ``quasi-Riesz transforms'' (at infinity), introduced by L. Chen in her PhD thesis. Our main result is an almost complete characterization of the sets of and such that the reverse quasi-Riesz inequality holds in the Vicsek cable system. It remains an open question to investigate reverse quasi-Riesz inequalities for other cable systems, or for manifolds built out of these
Reverse inequalities for quasi-Riesz transform on the Vicsek cable system
No full textThis work is devoted to the study of so-called ``reverse Riesz'' inequalities and suitable variants in the context of some fractal-like cable systems. It was already proved by L. Chen, T. Coulhon, J. Feneuil and the second author that, in the Vicsek cable system, the inequality is false for all . Following a recent joint paper by the two authors and M. Yang, we examine the validity of ``reverse quasi-Riesz'' inequalities, of the form , in the (unbounded) Vicsek cable system, for and . These reverse inequalities are strongly related to the problem of boundedness of the operators , the so-called ``quasi-Riesz transforms'' (at infinity), introduced by L. Chen in her PhD thesis. Our main result is an almost complete characterization of the sets of and such that the reverse quasi-Riesz inequality holds in the Vicsek cable system. It remains an open question to investigate reverse quasi-Riesz inequalities for other cable systems, or for manifolds built out of these
On the finiteness of the Morse Index for Schrödinger operators
Get PDF17 pagesLet H= be a Schrödinger on a complete non-compact manifold. It is known since the work of Fischer-Colbrie and Schoen that the finiteness of the negative spectrum of implies the existence of a function solution of outside a compact set. This has consequences for minimal surfaces and for the finiteness of spaces of harmonic sections in the Bochner method. Here we show that the converse statement also holds: if there exists solution of outside a compact set, then has a finite number of negative eigenvalues
A Gaussian estimate for the heat kernel on differential forms and application to the Riesz transform
Get PDF31 pagesLet be a m-dimensional complete Riemannian manifold which satisfies the n-Sobolev inequality and on which the volume growth is comparable to the one of for big balls; if the Hodge Laplacian on 1-forms is strongly positive and the Ricci tensor is in for an , then we prove a Gaussian estimate on the heat kernel of the Hodge Laplacian on 1-forms. This allows us to prove that, under the same hypotheses, the Riesz transform is bounded on for all $
A New Eigenvalue Problem for Free Boundary Minimal Submanifolds in the Unit Ball
No full textInternational audienc
A spectral result for Hardy inequalities
No full textTo appear in J. Math Pures Appl. This version: extensive changes in the exposition, among which: a new introduction, and a new section about spectrum and Agmon metrics. New results and examples have also been added.Let P be a linear, second order, elliptic operator satisfying a Hardy inequality with potential W (i.e. ) and best constant . We give conditions so that the spectrum of is . We apply this to several well-known Hardy inequalities: (improved) Hardy inequalities on a bounded convex domain with potential involving the distance to the boundary, and Hardy inequalities for minimal submanifolds of the Euclidean space
Opérateurs de Schrödinger et transformée de Riesz sur les variétés complètes non-compactes
Get PDFIn a first part, we give a necessary and sufficient condition so that a Schrödinger operator on a complete non-compact manifold has a finite number of negative eigenvalues. In a second part, we study the Riesz transform on a class of complete non-compact manifolds satisfying a Sobolev inequality. We first show a Gaussian estimate for the heat kernel of generalise Schrödinger operators, for example the Hodge Laplacian acting on differential forms, then we use this to show that the Riesz transform is bounded on the spaces for between and the Sobolev dimension. Finally, we show a perturbation result for the Riesz transform.Dans une première partie, on donne une condition nécessaire et suffisante à ce qu'un opérateur de Schrödinger sur une variété complète non-compacte ait un nombre fini de valeurs propres négatives. Dans une deuxième partie, on s'intéresse à la transformée de Riesz sur une classe de variétés complètes non-compactes vérifiant une inégalité de Sobolev. On montre d'abord une estimée gaussienne pour le noyau de la chaleur d'opérateurs de Schrödinger généralisés, comme par exemple le Laplacien de Hodge agissant sur les formes différentielles, puis on utilise ceci pour montrer que la transformée de Riesz est bornée sur les espaces si est compris entre et la dimension de Sobolev. Enfin, on montre un résultat de perturbation pour la transformée de Riesz
A Gaussian estimate for the heat kernel on differential forms and application to the Riesz transform
No full text31 pagesLet be a m-dimensional complete Riemannian manifold which satisfies the n-Sobolev inequality and on which the volume growth is comparable to the one of for big balls; if the Hodge Laplacian on 1-forms is strongly positive and the Ricci tensor is in for an , then we prove a Gaussian estimate on the heat kernel of the Hodge Laplacian on 1-forms. This allows us to prove that, under the same hypotheses, the Riesz transform is bounded on for all $
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