19,962 research outputs found
Contact equations, Lipschitz extensions and isoperimetric inequalities
We characterize locally Lipschitz mappings and existence of Lipschitz
extensions through a first order nonlinear system of PDEs. We extend this study
to graded group-valued Lipschitz mappings defined on compact Riemannian
manifolds. Through a simple application, we emphasize the connection between
these PDEs and the Rumin complex. We introduce a class of 2-step groups,
satisfying some abstract geometric conditions and we show that Lipschitz
mappings taking values in these groups and defined on subsets of the plane
admit Lipschitz extensions. We present several examples of these groups, called
Allcock groups, observing that their horizontal distribution may have any
codimesion. Finally, we show how these Lipschitz extensions theorems lead us to
quadratic isoperimetric inequalities in all Allcock groups.Comment: This version has additional references and a revisited introductio
A note on the Hanson-Wright inequality for random vectors with dependencies
We prove that quadratic forms in isotropic random vectors in
, possessing the convex concentration property with constant ,
satisfy the Hanson-Wright inequality with constant , where is an
absolute constant, thus eliminating the logarithmic (in the dimension) factors
in a recent estimate by Vu and Wang. We also show that the concentration
inequality for all Lipschitz functions implies a uniform version of the
Hanson-Wright inequality for suprema of quadratic forms (in the spirit of the
inequalities by Borell, Arcones-Gin\'e and Ledoux-Talagrand). Previous results
of this type relied on stronger isoperimetric properties of and in some
cases provided an upper bound on the deviations rather than a concentration
inequality.
In the last part of the paper we show that the uniform version of the
Hanson-Wright inequality for Gaussian vectors can be used to recover a recent
concentration inequality for empirical estimators of the covariance operator of
-valued Gaussian variables due to Koltchinskii and Lounici
Concentration of norms and eigenvalues of random matrices
We prove concentration results for operator norms of rectangular
random matrices and eigenvalues of self-adjoint random matrices. The random
matrices we consider have bounded entries which are independent, up to a
possible self-adjointness constraint. Our results are based on an isoperimetric
inequality for product spaces due to Talagrand.Comment: 15 pages; AMS-LaTeX; updated one referenc
The Kato Square Root Problem for Mixed Boundary Conditions
We consider the negative Laplacian subject to mixed boundary conditions on a
bounded domain. We prove under very general geometric assumptions that slightly
above the critical exponent its fractional power domains still
coincide with suitable Sobolev spaces of optimal regularity. In combination
with a reduction theorem recently obtained by the authors, this solves the Kato
Square Root Problem for elliptic second order operators and systems in
divergence form under the same geometric assumptions.Comment: Inconsistencies in Section 6 remove
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