Lp-Nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups

紧流形上傅里叶积分算子的核算子性、Schatten-von Neumann类、特征值分布与$L^p$-$L^q$有界性

Nuclearity, Schatten-von Neumann classes, distribution of eigenvalues and $L^p$-$L^q$-boundedness of Fourier integral operators on compact manifolds

本研究关联紧流形拉普拉斯特征函数Sogge型$L^p$估计与傅里叶积分算子$r$-核算子性判别，分析复/实典范关系下的算子类，给出Schatten-von Neumann类隶属条件、特征值衰减与迹的相关结果。

Approximation property and nuclearity on mixed-norm $L^p$, modulation and Wiener amalgam spaces

In this paper we first prove the metric approximation property for weighted mixed-norm $L_w^{(p_1,\dots ,p_n)}$ spaces. Using Gabor frame representation this implies that the same property holds in weighted modulation and Wiener amalgam spaces. As a consequence, Grothendieck's theory becomes applicable, and we give criteria for nuclearity and $r$-nuclearity for operators acting on these space as well as derive the corresponding trace formulae. Finally, we apply the notion of nuclearity to functions of the harmonic oscillator on modulation spaces.

Nuclear pseudo-differential operators in Besov spaces on compact Lie groups

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 multipliers on Besov spaces. Finally, we study the r-nuclearity, the Grothendieck-Lidskii formula and the (nuclear) trace of pseudo-differential operators in generalized Hörmander classes acting on periodic Besov spaces. We will restrict our attention to pseudo-differential operators with symbols of limited regularity.

Grothendieck's inequality in the noncommutative Schwartz space

In the spirit of Grothendieck's famous inequality from the theory of Banach spaces, we study a sequence of inequalities for the noncommutative Schwartz space, a Fréchet algebra of smooth operators. These hold in non-optimal form by a simple nuclearity argument. We obtain optimal versions and reformulate the inequalities in several different ways.

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 endowed with a positive measure, we introduce a notion of the operator's full symbol adapted to the Fourier analysis relative to a fixed elliptic operator. We give a description of Fourier multipliers, or of operators invariant relative to the elliptic operator. We apply these concepts to study Schatten classes of operators and to obtai…

Nuclear embeddings in weighted function spaces

We study nuclear embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weight belongs to some Muckenhoupt class and is essentially of polynomial type. Here we can extend our previous results [17,19] where we studied the compactness of corresponding embeddings. The concept of nuclearity goes back to Grothendieck who defined it in [14]. Recently there is a refreshed interest to study such questions [5-8,49]. This led us to the investigation in the weighted setting. We obtain complete characterisations for the nuclearity of the corresponding embedding. Essential tools are a …

Some Thoughts on Approximation Properties

We study some known approximation properties and introduce and investigate several new approximation properties, closely connected with different quasi-normed tensor products. These are the properties like the $AP_s$ or $AP_{(s,w)}$ for $s\in (0,1],$ which give us the possibility to identify the spaces of $s$-nuclear and $(s,w)$-nuclear operators with the corresponding tensor products (e.g., related to Lorentz sequence spaces). Some applications are given (in particular, we present not difficult proofs of the trace-formulas of Grothendieck-Lidskii type for several ideals of nuclear operators).

Nuclear embeddings of Morrey sequence spaces and smoothness Morrey spaces

We study nuclear embeddings for spaces of Morrey type, both in its sequence space version and as smoothness spaces of functions defined on a bounded domain $Ω\subset {\mathbb R}^d$. This covers, in particular, the meanwhile well-known and completely answered situation for spaces of Besov and Triebel-Lizorkin type defined on bounded domains which has been considered for a long time. The complete result was obtained only recently. Compact embeddings for function spaces of Morrey type have already been studied in detail, also concerning their entropy and approximation numbers. We now prove the fi…

Nuclear embeddings in general vector-valued sequence spaces with an application to Sobolev embeddings of function spaces on quasi-bounded domains

We study nuclear embeddings for spaces of Besov and Triebel-Lizorkin type defined on quasi-bounded domains $Ω\subset {\mathbb R}^d$. The counterpart for such function spaces defined on bounded domains has been considered for a long time and the complete answer was obtained only recently. Compact embeddings for function spaces defined on quasi-bounded domains have been studied in detail already, also concerning their entropy and $s$-numbers. We now prove the first and complete nuclearity result in this context. The concept of nuclearity has been introduce by Grothendieck in 1955 already. Our se…