175 research outputs found
Functional and isoperimetric inequalities for probability measures on H-type groups
We investigate isoperimetric and functional inequalities for probability measures in
the sub-elliptic setting and more specifically, on groups of Heisenberg type. The
approach we take is based on U-bounds as well as a Laplacian comparison theorem
for H-type groups. We derive different forms of functional inequalities (of [Phi]-entropy
and F-Sobolev type) and show that they can be equivalently stated as isoperimetric
inequalities at the level of sets. Furthermore, we study transportation of measure via
Talagrand-type inequalities. The methods used allow us to obtain gradient bounds for
the heat semigroup. Finally, we examine some properties of more general operators
given in Hormander’s sum of squares form and show that the associated semigroup
converges to a probability measure as t → [infinity]
Analysis of Schr\"odinger operators with inverse square potentials I: regularity results in 3D
Let be a potential on \RR^3 that is smooth everywhere except at a
discrete set \maS of points, where it has singularities of the form
, with for close to and continuous on
\RR^3 with for p \in \maS. Also assume that and
are smooth outside \maS and is smooth in polar coordinates around each
singular point. We either assume that is periodic or that the set \maS is
finite and extends to a smooth function on the radial compactification of
\RR^3 that is bounded outside a compact set containing \maS. In the
periodic case, we let be the periodicity lattice and define \TT :=
\RR^3/ \Lambda. We obtain regularity results in weighted Sobolev space for the
eigenfunctions of the Schr\"odinger-type operator acting on
L^2(\TT), as well as for the induced \vt k--Hamiltonians \Hk obtained by
restricting the action of to Bloch waves. Under some additional
assumptions, we extend these regularity and solvability results to the
non-periodic case. We sketch some applications to approximation of
eigenfunctions and eigenvalues that will be studied in more detail in a second
paper.Comment: 15 pages, to appear in Bull. Math. Soc. Sci. Math. Roumanie, vol. 55
(103), no. 2/201
Bounds on the non-real spectrum of differential operators with indefinite weights
Ordinary and partial differential operators with an indefinite weight
function can be viewed as bounded perturbations of non-negative operators in
Krein spaces. Under the assumption that 0 and are not singular
critical points of the unperturbed operator it is shown that a bounded additive
perturbation leads to an operator whose non-real spectrum is contained in a
compact set and with definite type real spectrum outside this set. The main
results are quantitative estimates for this set, which are applied to
Sturm-Liouville and second order elliptic partial differential operators with
indefinite weights on unbounded domains.Comment: 27 page
Gaussian Subordination for the Beurling-Selberg Extremal Problem
We determine extremal entire functions for the problem of majorizing,
minorizing, and approximating the Gaussian function by
entire functions of exponential type. This leads to the solution of analogous
extremal problems for a wide class of even functions that includes most of the
previously known examples (for instance \cite{CV2}, \cite{CV3}, \cite{GV} and
\cite{Lit}), plus a variety of new interesting functions such as
for ; \,, for
;\, ; and \,, for . Further applications to number theory include optimal
approximations of theta functions by trigonometric polynomials and optimal
bounds for certain Hilbert-type inequalities related to the discrete
Hardy-Littlewood-Sobolev inequality in dimension one
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