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Joachim Toft

Publications and source records attributed to Joachim Toft.

At least 37 records · Page 2Linked to original sources
arXiv2022

Wick and anti-Wick characterizations of linear operators on spaces of power series expansions

We study the link between Wick, anti-Wick and analytic kernel operators on the Bargmann transform side. We find classes of kernels, e.g. $\wideparen {\mathcal B} _s$, whose corresponding operators agree with the sets of linear and continuous operators on $\mathcal A _s$, the images of Pilipovi{ć} under the Bargmann transform. We show that in several situations, the sets of Wick, anti-Wick and kernel operators with symbols and kernels in $\wideparen {\mathcal B} _s$ agree. We also show some ring, module and composition properties for $\wideparen {\mathcal B} _s$, and similarly for other spaces related to $\wideparen {\mathcal B} _s$.

math.FA↗
arXiv2021

Pseudo-differential operators with isotropic symbols, Wick and anti-Wick operators, and hypoellipticity

We study the link between pseudo-differential operators and Wick operators via the Bargmann transform. We deduce a formula for the symbol of the Wick operator in terms of the short-time Fourier transform of the Weyl symbol. This gives characterizations of Wick symbols of pseudo-differential operators of Shubin type and of infinite order, and results on composition. We prove a series expansion of Wick operators in anti-Wick operators which leads to a sharp Gårding inequality and transition of hypoellipticity between Wick and and Shubin symbols. Finally we show continuity results for anti-Wick operators, and estimates for the Wick symbols of anti-Wick operators.

math.FA↗
arXiv2020

Translation and modulation invariant Hilbert spaces

We show that for any Hilbert space of distributions on $\textbf{R}^d$ which is translation and modulation invariant, is equal to $L^2(\textbf{R}^d)$, with the same norm apart from a multiplicative constant.

math.FA↗
arXiv2020

The Zak transform and Wiener estimates on Gelfand-Shilov and modulation spaces with applications to operator theory

We characterize Gelfand-Shilov spaces, their distribution spaces and modulation spaces in terms of estimates of their Zak transforms. We use these result for general investigations of quasi-periodic functions and distributions. We also establish necessary and sufficient conditions for linear operators in order for these operators should be conjugations by the Zak transform.

math.FA↗
arXiv2020

Characterizations of a class of Pilipovi{ć} spaces by powers of harmonic oscillator

We show that a smooth function $f$ on $\mathbf R^d$ belongs to the Pilipovi{ć} space $\mathcal H_{\flat _σ}(\mathbf R^d)$ or the Pilipovi{ć} space $\mathcal H_{0,\flat _σ}(\mathbf R^d)$, if and only if the $L^p$ norm of $H_d^Nf$ for $N\ge 0$, satisfy certain types of estimates. Here $H_d=|x|^2-Δ_x$ is the harmonic oscillator.

math.FA↗
arXiv2019

Paley-Wiener properties for spaces of power series expansions

We extend Paley-Wiener results in the Bargmann setting deduced earlier by the author together with E. Nabizadeh and C. Pfeuffer to larger class of power series expansions. At the same time we deduce characterisations of all Pilipovi{ć} spaces and their distributions (and not only of low orders as in the earlier work).

math.FA↗
arXiv2019

The pseudo-differential calculus in a Bargmann setting

We give a fundament for Berezin's analytic $Ψ$do considered in \cite{Berezin71} in terms of Bargmann images of Pilipovi{ć} spaces. We deduce basic continuity results for such $Ψ$do, especially when the operator kernels are in suitable mixed weighted Lebesgue spaces and act on certain weighted Lebesgue spaces of entire functions. In particular, we show how these results imply well-known continuity results for real $Ψ$do with symbols in modulation spaces, when acting on other modulation spaces.

math.FA↗
arXiv2019

Wiener estimates on modulation spaces

We characterise modulation spaces by suitable Wiener estimates on the short-time Fourier transforms of the involved functions and distributions. We use the results to refine some formulae on periodic distributions with Lebesgue estimates on their coefficients.

math.FA↗
arXiv2018

Tensor products for Gelfand-Shilov and Pilipovi{ć} distribution spaces

We show basic properties on tensor products for Gelfand-Shilov distributions and Pilipovi{ć} distributions. This also includes the Fubbini's property of such tensor products. We also apply the Fubbini property to deduce some properties for short-time Fourier transforms of Gelfand-Shilov and Pilipovi{ć} distributions.

math.FA↗
arXiv2018

Anisotropic Gevrey-Hörmander pseudo-differential operators on modulation spaces

We show continuity properties for the pseudo-differential operator $\operatorname{Op} (a)$ from $M(ω_0ω,\mathscr B )$ to $M(ω,\mathscr B )$, for fixed $s,σ\ge 1$, $ω,ω_0\in \mathscr P _{s,σ}^0$ ($ω,ω_0\in \mathscr P _{s,σ}$), $a\in Γ^{σ,s}_{(ω_0)}$ ($a\in Γ^{σ,s;0}_{(ω_0)}$) , and $\mathscr B$ is an invariant Banach function space.

math.FA↗
arXiv2018

Pseudo-differential calculus in anisotropic Gelfand-Shilov setting

We study some classes of pseudo-differential operators with symbols $a$ admitting anisotropic exponential growth at infinity and we prove mapping properties for these operators on Gelfand-Shilov spaces of type S. Moreover, we deduce algebraic and certain invariance properties of these classes.

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↗