Searcharxiv⌕ Search

arXiv subjects

Christine Pfeuffer

Publications and source records attributed to Christine Pfeuffer.

8 recordsLinked to original sources
arXiv2026

Compactness for pseudo-differential and Toeplitz operators on modulation spaces

We deduce various norm equivalences, and convolution estimates for the modulation space $M^{\sharp ,q}_{(\omega )}$ consisting of all $f\in M^{\infty ,q}_{(\omega )}$ such that $|V_\phi f \cdot \omega |$ satisfies a mild vanishing condition at infinity. We prove that $M^{\sharp ,q}_{(\omega )}$ is the completion of the Gelfand-Shilov space $\Sigma _1$ under the $M^{\infty ,q}_{(\omega )}$ norm. We use these results to deduce compactness for $\Psi$DO $\op (\mathfrak a )$, with $\mathfrak a \in M^{\sharp ,q}_{(\omega )}$, $0<q\le 1$, when acting on a broad family of modulation spaces.

math.FA↗
arXiv2024

Norm estimates for a broad class of modulation spaces, and continuity of Fourier type operators

Let $\mathscr B$ be a normal quasi-Banach function space with respect to $r_0 \in (0,1]$ and $v_0$, $\omega$ be $v$-moderate, and let $r\in [r_0,\infty ]$. Then we prove that $f$ belongs to the modulation space $M(\omega ,\mathscr B )$, iff $V_\phi f$ belongs to the Wiener amalgam space $W ^r(\omega ,\mathscr B )$, and $$ \| f \| _{M(\omega , \mathscr B)} \asymp \| V _\phi f \, \omega \| _{\mathscr B} \asymp \| V _\phi f\| _{W ^r(\omega, \mathscr B)}. $$ We also use the results to deduce continuity for pseudo-differential operators with symbols in weighted $M^{\infty,r_0}$-spaces, with $r_0\le 1$, when acting on $M(\omega ,\mathscr B )$-spaces.

math.FA↗
arXiv2020

Invariance of the Fredholm Index and Spectrum of Non-Smooth Pseudodifferential Operators

In this paper we show the invariance of the Fredholm index of non-smooth pseudodifferential operators with coefficients in Hölder spaces. By means of this invariance we improve previous spectral invariance results for non-smooth pseudodifferential operators $P$ with coefficients in Hölder spaces. For this purpose we approximate $P$ with smooth pseudodifferential operators and use a spectral invariance result of smooth pseudodifferential operators. Then we get the spectral invariance result in analogy to a proof of the spectral invariance result for non-smooth differential operators by Rabier.

math.FA↗
arXiv2018

Fredholm Property of Non-Smooth Pseudodifferential Operators

In this paper we prove sufficient conditions for the Fredholm property of a non-smooth pseudodifferential operator $P$ which symbol is in a Hölder space with respect to the spatial variable. As a main ingredient for the proof we use a suitable symbol-smoothing.

math.FA↗
arXiv2018

Compactness properties for modulation spaces

We prove that if $ω_1$ and $ω_2$ are moderate weights and $\mascB$ is a suitable (quasi-)Banach function space, then a necessary and sufficient condition for the embedding $i\, :\, M (ω_1,\mascB )\to M (ω_2,\mascB )$ between two modulation spaces to be compact is that the quotient $ω_2/ω_1$ vanishes at infinity. Moreover we show, that the boundedness of $ω_2/ω_1$ a necessary and sufficient condition for the previous embedding to be continuous.

math.FA↗
arXiv2016

Spectral Invariance of Non-Smooth Pseudodifferential Operators

In this paper we discuss some spectral invariance results for non-smooth pseudodifferential operators with coefficients in Hölder spaces. In analogy to the proof in the smooth case of Beals and Ueberberg, we use the characterization of non-smooth pseudodifferential operators to get such a result. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols and the fact, that in general the composition of two non-smooth pseudodifferential operators is not a pseudodifferential operator. In order to improve these spectral invariance results for certain subsets of non-smooth pseudodifferential operators with coefficients in Hölder spaces, we improve the characterization of non-smooth pseudodifferential operators in a previous work by the authors.

math.FA↗
arXiv2015

Characterization of Non-Smooth Pseudodifferential Operators

Smooth pseudodifferential operators on $\mathbb{R}^n$ can be characterized by their mapping properties between $L^p-$Sobolev spaces due to Beals and Ueberberg. In applications such a characterization would also be useful in the non-smooth case, for example to show the regularity of solutions of a partial differential equation. Therefore, we will show that every linear operator $P$, which satisfies some specific continuity assumptions, is a non-smooth pseudodifferential operator of the symbol-class $C^τ S^m_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols.

math.AP↗