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Spectral Theory, Function Spaces and Inequalities

Part of the Operator Theory: Advances and Applications Book Series (volume 219)

Editors: Brown, B. Malcolm; Lang, Jan; Wood, Ian G.

DOI: 10.1007/978-3-0348-0263-5

ISBN: 978-3-0348-0262-8

Publisher: Springer Basel - Springer Journals

Oct 11, 2011

1 - 10 of 14 Chapters

LitStream

Chapter 1Generalised Meissner Equations with an Eigenvalue-inducing Interface

Brown, B. M.; Eastham, M. S. P.

Pages: 1 –20

[An interface situation is considered, where a periodic differential equation is given on one side x > 0 of the interface and a general Sturm–Liouville equation is given on a finite interval (−X, 0) on the other side of the interface. A boundary condition is imposed at −X. The emphasis is on a...
LitStream

Chapter 2On the HELP Inequality for Hill Operators on Trees

Brown, B. M.; Schmidt, K. M.

Pages: 21 –36

[The validity of a generalised HELP inequality for a Schrödinger operator with periodic potential on a rooted homogeneous tree is related to the quasi-stability or quasi-instability of the associated differential equation. A numerical approach to the determination of the optimal constant in the...
LitStream

Chapter 3Measure of Non-compactness of Operators Interpolated by Limiting Real Methods

Cobos, Fernando; Fernández-Cabrera, Luz M.; Martínez, Antón

Pages: 37 –54

[We establish formulae for the measure of non-compactness of operators interpolated by limiting methods that come up by the choice θ = 0 and θ = 1 in the definition of the real method.]
LitStream

Chapter 4A New, Rearrangement-free Proof of the Sharp Hardy–Littlewood–Sobolev Inequality

Frank, Rupert L.; Lieb, Elliott H.

Pages: 55 –67

[We show that the sharp constant in the Hardy–Littlewood–Sobolev inequality can be derived using the method that we employed earlier for a similar inequality on the Heisenberg group. The merit of this proof is that it does not rely on rearrangement inequalities; it is the first one to do so for...
LitStream

Chapter 5Dichotomy in Muckenhoupt Weighted Function Space: A Fractal Example

Haroske, Dorothee D.

Pages: 69 –89

[We study dichotomy questions for weighted function spaces of Besov and Triebel–Lizorkin type where the weight is related to the distance of some point to a d-set Г, 0 < d < n, and the trace is taken on G. We can prove that – depending on the function space, the weight and the set Г – there...
LitStream

Chapter 6Lavrentiev’s Theorem and Error Estimation in Elliptic Inverse Problems

Knowles, Ian; LaRussa, Mary A.

Pages: 91 –103

[Lavrentiev’s theorem provides bounds for analytic functions known to be small at a finite number of points in a bounded region. An analogous result is established for solutions of elliptic equations on bounded regions in ℝ2 and applied to estimating non-uniqueness error in elliptic inverse...
LitStream

Chapter 7Two-weighted Norm Inequalities for the Double Hardy Transforms and Strong Fractional Maximal Functions in Variable Exponent Lebesgue Space

Kokilashvili, Vakhtang; Meskhi, Alexander

Pages: 105 –124

[Two-weight norm estimates for double Hardy transforms and variable order strong fractional maximal functions are established in variable exponent Lebesgue spaces. Derived conditions are simultaneously necessary and sufficient in the case when the exponent function of the right–hand side space is...
LitStream

Chapter 8Modular Eigenvalues of the Dirichlet p(·)-Laplacian and Their Stability

Lang, Jan; Méndez, Osvaldo

Pages: 125 –137

[The concept of the modular first Dirichlet eigenvalue for the p(·)-Laplacian is introduced as a generalization of the constant case. An important property of the corresponding eigenfunctions is obtained. We prove a qualitative stability result for such eigenvalues in terms of the magnitude of...
LitStream

Chapter 9Spectral Properties of Some Degenerate Elliptic Differential Operators

Lewis, Roger T.

Pages: 139 –156

[In this paper we extend classical criteria for determining lower bounds for the least point of the essential spectrum of second-order elliptic differential operators on domains Ω ⊂ ℝn allowing for degeneracy of the coefficients on the boundary. We assume that we are given a sesquilinear form and...
LitStream

Chapter 10Continuous and Compact Embeddings of Bessel-Potential-Type Spaces

Opic, Bohumír

Pages: 157 –196

[We survey, comment and complement our results on embeddings of Bessel potential spaces modelled upon Lorentz–Karamata spaces. Target spaces in our embeddings are either Lorentz–Karamata spaces or generalized Hölder spaces. In particular, we establish necessary and sufficient conditions both for...

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