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Bojan Prangoski

Publications and source records attributed to Bojan Prangoski.

At least 19 recordsLinked to original sources
arXiv2026

On the pullback on spaces of distributions with prescribed microregularity with respect to a general Banach space

We consider the space $\mathcal{D}'^E_L(U)$ of all distributions on the open set $U$ whose wave front set measured with respect to a Banach space $E$ lies in a closed conic subset $L$ of $U\times(\mathbb{R}^n\backslash\{0\})$. The Banach space $E$ only satisfies mild technical assumptions. We show that the pullback by a diffeomorphism $f:O\rightarrow U$ is a topological isomorphism $f^*:\mathcal{D}'^E_L(U)\rightarrow\mathcal{D}'^E_{f^*L}(O)$. In the case when $E$ is the $L^p$-Sobolev space $W^{r,p}(\mathbb{R}^n)$ of order $r\in\mathbb{R}$, we study the pullback by a smooth map $f:O\rightarrow U$ of constant rank.

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arXiv2025

On a class of Mikhlin multipliers which do not preserve $L^1$-, $L^\infty$-regularity and continuity

We show that every Fourier multiplier with real-valued and positively homogeneous symbol of order 0, supported in a cone whose dual cone has a nonempty interior and such that the average of the positive part is sufficiently larger than the average of the negative part does not preserve the $L^1$- nor the $L^\infty$ regularity and neither the continuity.We also construct wave front sets which measure the microlocal regularity with respect to a large class of Banach spaces. As a consequence of the first part, we argue that one can never construct wave front sets that behave in a natural way and measure the microlocal $L^1$- nor $L^\infty$-regularity and neither the continuity

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arXiv2024

Spaces of distributions with Sobolev wave front in a fixed conic set: compactness, pullback by smooth maps and the compensated compactness theorem

We consider the space $\mathcal{D}'^r_L(M;E)$ of distributional sections of the smooth complex vector bundle $E\rightarrow M$ whose Sobolev wave front set of order $r\in\mathbb{R}$ lies in the closed conic subset $L$ of $T^*M\backslash0$. We introduce a locally convex topology on it to study the continuity of the pullback by smooth maps and generalise the result of Hörmander about the pullback on the space of distributions with $\mathcal{C}^{\infty}$ wave front set in $L$. We employ an idea of Gérard [18] to extend the Kolmogorov-Riesz compactness theorem to $\mathcal{D}'^r_L(M;E)$ and we characterise its relatively compact subsets. We study the continuity properties of pseudo-differential operators when acting on $\mathcal{D}'^r_L(M;E)$, $r\in\mathbb{R}$, and we generalise the Rellich's lemma. As an application of our results, we extend the microlocal defect measures of Gérard and Tartar to sequences in $\mathcal{D}'^0_L(M;E)$ and we show a microlocal variant of the compensated compactness theorem.

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arXiv2024

Extension of Localisation Operators to Ultradistributional Symbols With Super-Exponential Growth

In the Gelfand-Shilov setting, the localisation operator $A^{φ_1,φ_2}_a$ is equal to the Weyl operator whose symbol is the convolution of $a$ with the Wigner transform of the windows $φ_2$ and $φ_1$. We employ this fact, to extend the definition of localisation operators to symbols $a$ having very fast super-exponential growth by allowing them to be mappings from ${\mathcal D}^{\{M_p\}}(\mathbb R^d)$ into ${\mathcal D}'^{\{M_p\}}(\mathbb R^d)$, where $M_p$, $p\in\mathbb N$, is a non-quasi-analytic Gevrey type sequence. By choosing the windows $φ_1$ and $φ_2$ appropriately, our main results show that one can consider symbols with growth in position space of the form $\exp(\exp(l|\cdot|^q))$, $l,q>0$.

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arXiv2022

Wiener amalgam spaces of quasianalytic ultradistributions

We define Wiener amalgam spaces of (quasi)analytic ultradistributions whose local components belong to a general class of translation and modulation invariant Banach spaces of ultradistributions and their global components are either weighted $L^p$ or weighted $\mathcal{C}_0$ spaces. We provide a discrete characterisation via so called uniformly concentrated partitions of unity. Finally, we study the complex interpolation method and we identify the strong duals for most of these Wiener amalgam spaces.

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arXiv2022

Characterisation of the Weyl-Hörmander classes by time-frequency shifts

We characterise the Weyl-Hörmander symbol classes $S(M,g)$ via the growth of the action of the corresponding $Ψ$DOs on time-frequency shifts of a single test function. For this purpose, we introduce a geometric short-time Fourier transform which is well-suited for the analysis of $S(M,g)$. We define new modulation spaces and achieve the characterisation of the Weyl-Hörmander classes by showing that they are intersections of such modulation spaces suitable for the time-frequency characterisation.

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arXiv2021

Infinite order $Ψ$DOs: Composition with entire functions, new Shubin-Sobolev spaces, and index theorem

We study global regularity and spectral properties of power series of the Weyl quantisation $a^w$, where $a(x,ξ) $ is a classical elliptic Shubin polynomial. For a suitable entire function $P$, we associate two natural infinite order operators to $a^{w}$, $P(a^w)$ and $(P\circ a)^{w},$ and prove that these operators and their lower order perturbations are globally Gelfand-Shilov regular. They have spectra consisting of real isolated eigenvalues diverging to $\infty$ for which we find the asymptotic behaviour of their eigenvalue counting function. In the second part of the article, we introduce Shubin-Sobolev type spaces by means of $f$-$Γ^{*,\infty}_{A_p,ρ}$-elliptic symbols, where $f $ is a function of ultrapolynomial growth and $Γ^{*,\infty}_{A_p,ρ}$ is a class of symbols of infinite order studied in this and our previous papers. We study the regularity properties of these spaces, and show that the pseudo-differential operators under consideration are Fredholm operators on them. Their indices are independent on the order of the Shubin-Sobolev spaces; finally, we show that the index can be expressed via a Fedosov-Hörmander integral formula.

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arXiv2021

Gabor frame characterisations of generalised modulation spaces

We obtain Gabor frame characterisations of modulation spaces defined via a class of translation-modulation invariant Banach spaces of distributions that was recently introduced in $[10]$. We show that these spaces admit an atomic decomposition through Gabor expansions and that they are characterised by summability properties of their Gabor coefficients. Furthermore, we construct a large space of admissible windows. This generalises several fundamental results for the classical modulation spaces $ M^{p,q}_{w}$. Due to the absence of solidity assumptions on the Banach spaces defining these modulation spaces, the methods used for the spaces $M^{p,q}_{w}$ (or, more generally, in coorbit space theory) fail in our setting and we develop here a new approach based on the twisted convolution.

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arXiv2020

Modulation spaces associated to tensor products of amalgam spaces

We identify the modulation spaces associated to tensor products of amalgam spaces having a large class of Banach spaces as their local component. As consequences of the main results, we describe the modulation spaces associated to tensor products of various $L^p$ spaces.

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arXiv2020

Factorization in Denjoy-Carleman classes associated to representations of $(\mathbb{R}^{d},+)$

For two types of moderate growth representations of $(\mathbb{R}^d,+)$ on sequentially complete locally convex Hausdorff spaces (including F-representations [J. Funct. Anal. 262 (2012), 667-681], we introduce Denjoy-Carleman classes of ultradifferentiable vectors and show a strong factorization theorem of Dixmier-Malliavin type for them. In particular, our factorization theorem solves [Conjecture 6.; J. Funct. Anal. 262 (2012), 667-681] for analytic vectors of representations of $G =(\mathbb{R}^d,+)$. As an application, we show that various convolution algebras and modules of ultradifferentiable functions satisfy the strong factorization property.

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arXiv2019

On the projective description of spaces of ultradifferentiable functions of Roumieu type

We provide a projective description of the space $\mathcal{E}^{\{\mathfrak{M}\}}(Ω)$ of ultradifferentiable functions of Roumieu type, where $Ω$ is an arbitrary open set in $\mathbb{R}^d$ and $\mathfrak{M}$ is a weight matrix satisfying the analogue of Komatsu's condition $(M.2)'$. In particular, we obtain in a unified way projective descriptions of ultradifferentiable classes defined via a single weight sequence (Denjoy-Carleman approach) and via a weight function (Braun-Meise-Taylor approach) under considerably weaker assumptions than in earlier versions of these results.

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arXiv2019

Equivalence of Ellipticity and Fredholmness in the Weyl-Hörmander calculus

The main result is that the Fredholm property of a $Ψ$DO acting on Sobolev spaces in the Weyl-Hörmander calculus and the ellipticity are equivalent for geodesically temperate Hörmanders metrics whose associated Planck's functions vanish at infinity. Additionally, we prove that when the Hörmander metric is geodesically temperate, and consequently the calculus is spectrally invariant, the inverse $λ\mapsto b_λ\in S(1,g)$ of every $\mathcal{C}^N$, $0\leq N\leq \infty$, map $λ\mapsto a_λ\in S(1,g)$ comprised of invertible elements on $L^2$ is again of class $\mathcal{C}^N$.

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arXiv2018

Translation-modulation invariant Banach spaces of ultradistributions

We introduce and study a new class of translation-modulation invariant Banach spaces of ultradistributions. These spaces show stability under Fourier transform and tensor products; furthermore, they have a natural Banach convolution module structure over a certain associated Beurling algebra, as well as a Banach multiplication module structure over an associated Wiener-Beurling algebra. We also investigate a new class of modulation spaces, the Banach spaces of ultradistributions $\mathcal{M}^F$ on $\mathbb{R}^{d}$, associated to translation-modulation invariant Banach spaces of ultradistributions $F$ on $\mathbb{R}^{2d}$.

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arXiv2018

Spectral asymptotics for infinite order pseudo-differential operators

We study spectral properties of a class of global infinite order pseudo-differential operators and obtain the asymptotic behaviour of the spectral counting functions of such operators. Unlike their finite order counterparts, their spectral asymptotics are not of power-log-type but of log-type. The ultradistributional setting of such operators of infinite order makes the theory more complex so that the standard finite order global Weyl calculus cannot be used in this context.

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