# Grothendieck-Lidskii trace formula for mixed-norm and variable Lebesgue spaces

@article{Delgado2016GrothendieckLidskiiTF, title={Grothendieck-Lidskii trace formula for mixed-norm and variable Lebesgue spaces}, author={J. Delgado and Michael Ruzhansky and Baoxiang Wang}, journal={arXiv: Functional Analysis}, year={2016} }

In this note we present the metric approximation property for weighted mixed-norm $L_w^{(p_1,\dots ,p_n)}$ and variable exponent Lebesgue type spaces. As a consequence, this also implies the same property for modulation and Wiener-Amalgam spaces. We then characterise nuclear operators on such spaces and state the corresponding Grothendieck-Lidskii trace formulae. We apply the obtained results to derive criteria for nuclearity and trace formulae for periodic operators on $\mathbb R^n$ and…

## 12 Citations

The bounded approximation property of variable Lebesgue spaces and nuclearity

- Mathematics
- 2015

In this paper we prove the bounded approximation property for variable exponent Lebesgue spaces, study the concept of nuclearity on such spaces and apply it to trace formulae such as the…

Schatten properties, nuclearity and minimality of phase shift invariant spaces

- MathematicsApplied and Computational Harmonic Analysis
- 2019

We extend Feichtinger's minimality property on smallest non-trivial time-frequency shift invariant Banach spaces, to the quasi-Banach case. Analogous properties are deduced for certain matrix…

Nuclear Pseudo-Differential Operators in Besov Spaces on Compact Lie Groups

- Mathematics
- 2016

In this work we establish the metric approximation property for Besov spaces defined on arbitrary compact Lie groups. As a consequence of this fact, we investigate trace formulae for nuclear Fourier…

On the nuclear trace of Fourier integral operators

- MathematicsRevista Integración
- 2019

In this paper we characterise ther-nuclearity of Fourier integraloperators on Lebesgue spaces. Fourier integral operators will be consideredinRn,the discrete groupZn,then-dimensional torus and…

On Function Spaces with Mixed Norms --- A Survey

- Mathematics
- 2019

The targets of this article are threefold. The first one is to give a survey on the recent developments of function spaces with mixed norms, including mixed Lebesgue spaces, iterated weak Lebesgue…

On quasianalytic classes of Gelfand–Shilov type. Parametrix and convolution

- MathematicsJournal de Mathématiques Pures et Appliquées
- 2018

We develop a convolution theory for quasianalytic ultradistributions of Gelfand-Shilov type. We also construct a special class of ultrapolynomials, and use it as a base for the parametrix method in…

Nuclearity for Fourier integral operators in $L^p$-spaces

- Mathematics
- 2018

In this note we study sharp sufficient conditions for the nuclearity of Fourier integral operators on $L^p$-spaces, $1< p\leq 2$. Our conditions and those presented in Cardona [2] provide a…

A brief description of operators associated to the quantum harmonic oscillator on Schatten-von Neumann classes

- Physics, MathematicsRevista Integración
- 2018

In this note we study pseudo-multipliers associated to the harmonic oscillator (also called Hermite multipliers) belonging to Schatten classes on $L^2(\mathbb{R}^n)$. We also investigate the spectral…

$$L^p$$Lp-Boundedness and $$L^p$$Lp-Nuclearity of Multilinear Pseudo-differential Operators on $${\mathbb {Z}}^n$$Zn and the Torus $${\mathbb {T}}^n$$Tn

- MathematicsJournal of Fourier Analysis and Applications
- 2019

In this article, we begin a systematic study of the boundedness and the nuclearity properties of multilinear periodic pseudo-differential operators and multilinear discrete pseudo-differential…

A brief description of operators associated to the quantum harmonic oscillator on Schatten-von Neumann classes

- 2018

In this note we study pseudo-multipliers associated to the harmonic oscillator (also called Hermite multipliers) belonging to Schatten classes on L(R). We also investigate the spectral trace of these…

## References

SHOWING 1-10 OF 24 REFERENCES

Approximation property and nuclearity on mixed-norm Lp, modulation and Wiener amalgam spaces

- Mathematics, Computer ScienceJ. Lond. Math. Soc.
- 2016

The metric approximation property for weighted mixed-norm L (p1,...,pn) w spaces is proved and Grothendieck’s theory becomes applicable, and the notion of nuclearity is applied to functions of the harmonic oscillator on modulation spaces.

The bounded approximation property of variable Lebesgue spaces and nuclearity

- Mathematics
- 2015

In this paper we prove the bounded approximation property for variable exponent Lebesgue spaces, study the concept of nuclearity on such spaces and apply it to trace formulae such as the…

Lebesgue and Sobolev Spaces with Variable Exponents

- Mathematics
- 2011

1 Introduction.- 2 A framework for function spaces.- 3 Variable exponent Lebesgue spaces.- 4 The maximal operator.- 5 The generalized Muckenhoupt condition*.- 6 Classical operators.- 7 Transfer…

Changes of variables in modulation and Wiener amalgam spaces

- Mathematics
- 2008

In this paper various properties of global and local changes of variables as well as properties of canonical transforms are investigated on modulation and Wiener amalgam spaces. We establish several…

Variable Lebesgue Spaces and Hyperbolic Systems

- Mathematics
- 2014

Part I: Introduction to the Variable Lebesgue Spaces.- Introduction and motivation.- Properties of variable Lebesgue spaces.- The Hardy-Littlewood maximal operator.- Extrapolation in variable…

Quantization of Pseudo-differential Operators on the Torus

- Mathematics
- 2008

AbstractPseudo-differential and Fourier series operators on the torus
${{\mathbb{T}}^{n}}=(\Bbb{R}/2\pi\Bbb{Z})^{n}$
are analyzed by using global representations by Fourier series instead of local…

Absolutely summing operators on C[0,1] as a tree space and the bounded approximation property☆

- Mathematics
- 2010

Let X be a Banach space. For describing the space P(C[0,1],X) of absolutely summing operators from C[0,1] to X in terms of the space X itself, we construct a tree space l1tree(X) on X. It consists of…

Pseudo-Differential Operators and Symmetries: Background Analysis and Advanced Topics

- Mathematics
- 2009

Preface.- Introduction.- Part I Foundations of Analysis.- A Sets, Topology and Metrics.- B Elementary Functional Analysis.- C Measure Theory and Integration.- D Algebras.- Part II Commutative…

Banach spaces related to integrable group representations and their atomic decompositions, I

- Mathematics
- 1989

Abstract We present a general theory of Banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states,…

Navier–Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices

- Mathematics
- 2010

Abstract The Cauchy problems for Navier–Stokes equations and nonlinear heat equations are studied in modulation spaces M q , σ s ( R n ) . Though the case of the derivative index s = 0 has been…