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Titchmarsh theorems for Fourier transforms of Hölder–Lipschitz functions on compact homogeneous manifolds

 Compact Lie groups Lipschitz functions  Communicated by A. Constantin. Julio Delgado was supported by the Leverhulme Research Grant RPG-2017-151. Michael Ruzhansky was supported by the EPSRC Grant EP

Fourier multipliers, symbols, and nuclearity on compact manifolds

The notion of invariant operators, or Fourier multipliers, is discussed for densely defined operators on Hilbert spaces, with respect to a fixed partition of the space into a direct sum of finite-dimensional subspaces. As a consequence, given a compact manifold M endowed with a positive measure, we introduce a notion of the operator’s full symbol adapted to the Fourier analysis...

Smooth Dense Subalgebras and Fourier Multipliers on Compact Quantum Groups

We define and study dense Frechet subalgebras of compact quantum groups realised as smooth domains associated with a Dirac type operator with compact resolvent. Further, we construct spectral triples on compact matrix quantum groups in terms of Clebsch–Gordon coefficients and the eigenvalues of the Dirac operator \({\mathcal{D}}\). Grotendieck’s theory of topological tensor...

Weighted \(L^{p}\) -Hardy and \(L^{p}\) -Rellich inequalities with boundary terms on stratified Lie groups

In this paper, generalised weighted \(L^p\)-Hardy, \(L^p\)-Caffarelli–Kohn–Nirenberg, and \(L^p\)-Rellich inequalities with boundary terms are obtained on stratified Lie groups. As consequences, most of the Hardy type inequalities and Heisenberg–Pauli–Weyl type uncertainty principles on stratified groups are recovered. Moreover, a weighted \(L^2\)-Rellich type inequality with the...

Convolution, Fourier analysis, and distributions generated by Riesz bases

In this note we discuss notions of convolutions generated by biorthogonal systems of elements of a Hilbert space. We develop the associated biorthogonal Fourier analysis and the theory of distributions, discuss properties of convolutions and give a number of examples.

Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness

Michael Ruzhansky 0 1 0 Department of Mathematics, Imperial College London, 180 Queen's Gate , London SW7 2AZ , UK 1 Department of Mathematical Sciences, Loughborough University , Loughborough ... can be applied. We give explicit details for the appearing Michael Ruzhansky was supported in parts by EPSRC Grant EP/R003025/1 and by the Leverhulme Grant RPG-2017-151. No new data was collected or

Sobolev Type Inequalities, Euler–Hilbert–Sobolev and Sobolev–Lorentz–Zygmund Spaces on Homogeneous Groups

. Michael Ruzhansky (B) and Nurgissa Yessirkegenov Department of Mathematics Imperial College London 180 Queen’s Gate London SW7 2AZ UK e-mail: Durvudkhan Suragan and Nurgissa Yessirkegenov Institute of

Wave Equation for Operators with Discrete Spectrum and Irregular Propagation Speed

. . . . . . . . . . . . . . . . . . . . . . . 3.4. Regular Elliptic Boundary Value Problems . . . . . . . . . . . . . . . . . . Michael Ruzhansky was supported in parts by the EPSRC Grant EP/K039407/1 and by the Leverhulme Grant RPG-2014-02. Niyaz

Erratum to “The Gohberg Lemma, compactness, and essential spectrum of operators on compact Lie groups”


Caffarelli–Kohn–Nirenberg and Sobolev type inequalities on stratified Lie groups

In this short paper, we establish a range of Caffarelli–Kohn–Nirenberg and weighted \(L^{p}\)-Sobolev type inequalities on stratified Lie groups. All inequalities are obtained with sharp constants. Moreover, the equivalence of the Sobolev type inequality and Hardy inequality is shown in the \(L^{2}\)-case.

On \(C^\infty \) well-posedness of hyperbolic systems with multiplicities

In this paper, we study first-order hyperbolic systems of any order with multiple characteristics (weakly hyperbolic) and time-dependent analytic coefficients. The main question is when the Cauchy problem for such systems is well-posed in \(C^{\infty }\) and in \({\mathcal {D}}'\). We prove that the analyticity of the coefficients combined with suitable hypotheses on the...

Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field

/K039407/1 and by the Leverhulme Grant RPG-2014-02, as well as by the MESRK Grant 0773/GF4. No new data were collected or generated during the course of research. B Michael Ruzhansky 1 Introduction The

Smoothing estimates for non-dispersive equations

JSPS KAKENHI 26287022 and 26610021. B Michael Ruzhansky which is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator a(Dx

On hyperbolic systems with time-dependent Hölder characteristics

In this paper, we study the well-posedness of weakly hyperbolic systems with time-dependent coefficients. We assume that the eigenvalues are low regular, in the sense that they are Hölder with respect to t. In the past, these kinds of systems have been investigated by Yuzawa (J Differ Equ 219(2):363–374, 2005) and Kajitani and Yuzawa (Ann Sc Norm Super Pisa Cl Sci (5) 5(4):465...

On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries

In this note we prove an analogue of the Rayleigh–Faber–Krahn inequality, that is, that the geodesic ball is a maximiser of the first eigenvalue of some convolution type integral operators, on the sphere \(\mathbb {S}^{n}\) and on the real hyperbolic space \(\mathbb {H}^{n}\). It completes the study of such question for complete, connected, simply connected Riemannian manifolds...

\(L^p\) Fourier multipliers on compact Lie groups

Mathematische Zeitschrift Michael Ruzhansky 0 Jens Wirth 0 Mathematics Subject Classification 0 J. Wirth Institut fur Analysis, Dynamik und Modellierung, Universitat Stuttgart , Pfaffenwaldring

Hyperbolic Second Order Equations with Non-Regular Time Dependent Coefficients

In this paper we study weakly hyperbolic second order equations with time dependent irregular coefficients. This means assuming that the coefficients are less regular than Hölder. The characteristic roots are also allowed to have multiplicities. For such equations, we describe the notion of a ‘very weak solution’ adapted to the type of solutions that exist for regular...

On global inversion of homogeneous maps

Michael Ruzhansky Mitsuru Sugimoto Mathematics Subject Classification In this note we prove a global inverse function theorem for homogeneous mappings on Rn. The proof is based on an adaptation of

Hörmander Class of Pseudo-Differential Operators on Compact Lie Groups and Global Hypoellipticity

In this paper we give several global characterisations of the Hörmander class \(\Psi ^m(G)\) of pseudo-differential operators on compact Lie groups in terms of the representation theory of the group. The result is applied to give criteria for the ellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several...

Weakly hyperbolic equations with non-analytic coefficients and lower order terms

Claudia Garetto 0 Michael Ruzhansky 0 Mathematics Subject Classification 0 0 C. Garetto School of Mathematics, Loughborough University , Leicestershire LE11 3TU, UK In this paper we consider