Sharp embeddings between weighted Paley-Wiener spaces

In this paper we address the problem of estimating the operator norm of the embeddings between multidimensional weighted Paley-Wiener spaces. These can be equivalently thought as Fourier uncertainty principles for bandlimited functions. By means of radial symmetrization mechanisms, we show that such problems can all be shifted to dimension one. We provide precise asymptotics in the general case and, in some particular situations, we are able to identify the sharp constants and characterize the extremizers. The sharp constant study is actually a consequence of a more general result we prove in the setup of de Branges spaces of entire functions, addressing the operator given by multiplication by $z^k$, $k \in \mathbb{N}$. Applications to sharp higher order Poincar\'{e} inequalities and other related extremal problems are discussed.Comment: 42 page

Multipliers in Hardy and Bergman Spaces, and Riesz Decomposition

Multipliers' methods have proven to be an efficient tool in virtually any area of Analysis. Many linear operators act as multipliers on Taylor series, Fourier series, Fourier integrals, etc., of a function. This means the operators introduce some multiplicative factors to the series or integrals. As a consequence, conditions on boundedness of multipliers imply important inequalities in Analysis, in particular, in Approximation Theory.We consider series multipliers in Hardy and Bergman spaces in the unit disk D of the complex plane C, as well as multipliers of Fourier integrals in Hardy spaces in tubes over open cones (in C^n). Obtained conditions are used to derive some inequalities, e.g., Bernstein and Nikolskii type inequalities for entire functions.Some of the multiplier conditions are surprisingly sharp. As an example, a critical index for Bochner-Riesz means of Fourier integrals in Hardy spaces in tubes has been found.For the Hadamard product of two polynomials (again, a multiplier-type operator), we obtain sharp inequalities for its Mahler measure. They imply several sharp inequalities used in Approximation Theory.We conclude the thesis by the Riesz Decomposition result for m-superharmonic functions in R^n, (2m is strictly less than n), which generalizes work of K. Kitaura and Y. Mizuta for super-biharmonic functions.Mathematic

3D-Rekonstruktion eines bronzezeitlichen Ägäisschiffes

Paper I. Trace Formulae and Spectral Properties of FourthOrder Differential Operators.We derive trace formulae forfourth order differential operators in dimension one anddiscuss their connection with sharp Lieb-Thirring inequalitiesfor the Riesz means of negative eigenvalues of order γ≥ 7/4. We also construct reflectionless potentials forfourth order differential operators. Paper II. Lieb-Thirring Inequalities for Higher OrderDifferential Operators.We derive Lieb-Thirringinequalities for the Riesz means of eigenvalues of order γ≥ 3=4 for a fourth order operator in arbitrarydimensions. We also consider some extensions to polyharmonicoperators, and to systems of such operators, in dimensionsgreaterthan one. Paper III. Follytons and the Removal of Eigenvalues forFourth Order Differential Operators.(Joint with J. Hoppeand A. Laptev). A non-linear functional Q[u, v]is given that governs the loss, respectively gain,of (doubly degenerate) eigenvalues of fourth order differentialoperators L = ∂4+ ∂ u ∂ + v on the line. Apart fromfactorizing L as A*A + E0, providing several explicit examples, and derivingvarious relations betweenu, vand eigenfunctions of L, we finduandvsuch that L is isospectral to the free operator L0= ∂4 up to one (multiplicity 2) eigenvalueE0&lt;0. Not unexpectedly, this choice ofu, vleads to exact solutions of the correspondingtime-dependent PDE\u92s.QC 20161027</p

Eigenvalue estimates for Schroedinger operators on metric trees

We consider Schroedinger operators on regular metric trees and prove Lieb-Thirring and Cwikel-Lieb-Rozenblum inequalities for their negative eigenvalues. The validity of these inequalities depends on the volume growth of the tree. We show that the bounds are valid in the endpoint case and reflect the correct order in the weak or strong coupling limit