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58

Jun 30, 2018
06/18

by
Charles Batty; Alexander Borichev; Yuri Tomilov

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We prove $L^p$-analogues of the classical tauberian theorem of Ingham and Karamata, and its variations giving rates of decay. These results are applied to derive $L^p$-decay of operator families arising in the study of the decay of energy for damped wave equations and local energy for wave equations in exterior domains. By constructing some examples of critical behaviour we show that the $L^p$-rates of decay obtained in this way are best possible under our assumptions.

Topics: Complex Variables, Functional Analysis, Mathematics, Analysis of PDEs, Dynamical Systems

Source: http://arxiv.org/abs/1403.6084

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37

Sep 19, 2013
09/13

by
Alexander Borichev; Hakan Hedenmalm; Alexander Volberg

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In a wide class of weighted Bergman spaces, we construct invertible non-cyclic elements. These are then used to produce z-invariant subspaces of index higher than one. In addition, these elements generate nontrivial bilaterally invariant subspaces in anti-symmetrically weighted Hilbert spaces of sequences.

Source: http://arxiv.org/abs/math/0304318v1

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68

Sep 20, 2013
09/13

by
Alexander Borichev; Fedor Nazarov; Mikhail Sodin

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Let F be a class of functions with the uniqueness property: if a function f in F vanishes on a set of positive measure, then f is the zero function. In many instances, we would like to have a quantitative version of this property, e.g. a lower bound for |f| outside a small exceptional set. Such estimates are well-known and useful for polynomials, complex- and real-analytic functions, exponential polynomials. In this work we prove similar results for the Denjoy-Carleman and the Bernstein classes...

Source: http://arxiv.org/abs/math/0301217v2