arXiv subjects

Stevan Pilipović

Publications and source records attributed to Stevan Pilipović.

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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arXiv2026

A parametrix construction for time-fractional partial differential equations

We prove in detail how to construct the parametrix of a parameter-dependent family of pseudodifferential operators, appearing in the analysis of time-fractional partial differential equations. In particular, we perform a precise study of the dependence from the parameter of the corresponding asymptotic expansion terms, as well as of the smoothing remainders. Moreover, we provide some results about the Laplace transform of vector-valued distributions, also appearing in the analysis of time-fractional partial differential equations.

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arXiv2026

Representation formula, regularity and decay of solutions for sub-diffusion type equations

We study regularity and decay properties for the solutions of the Cauchy problem for time-fractional partial differential equations, with tempered initial data, belonging to suitable (weighted) Sobolev spaces, associated with a differential operator on space variables with polynomially bounded coefficients. We obtain a representation formula for the solution, modulo time-regular functions, smooth and rapidly decreasing with respect to the space variables. By means of the representation formula, the (decay and smoothness) singularities of the solution of the homogeneous Cauchy problem can be controlled, in terms of (global) wavefront sets of the initial data.

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arXiv2025

Abelian and Tauberian results for the fractional Hankel transform in Zemanian-type spaces

In this paper, we first present an Abelian-type theorem for the fractional Hankel transform (FrHT) within Zemanian generalized function spaces. To prove this, we show that these spaces have the Montel property. Next, we construct a new Zemanian-type space as a projective limit of suitable Banach spaces. Its dual is the largest known distribution space admitting the FrHT. Finally, within this extended setting, we establish new Abelian and Tauberian-type results for the FrHT.

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arXiv2025

Abelian and Tauberian results for the fractional Hankel transform in Zemanian-type spaces

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arXiv2025

Convolution and product in finitely generated shift-invariant spaces

We discuss some structural properties of finitely generated shift-invariant (FGSI) spaces and subspaces of Sobolev spaces, particularly those related to convolution and the product within these spaces. We find shift-invariant solutions in FGSI spaces for a class of differential-difference equations with constant coefficients. Additionally, we analyze the Fourier multipliers in FGSI spaces and the wave fronts for the convolution and product in FGSI spaces.

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arXiv2025

Abelian and Tauberian Results for the Fractional Hankel Transform of Generalized Functions

This paper aims to explore the quasiasymptotic behavior of distributions through the fractional Hankel transform. We present Tauberian result that connects the asymptotic behavior of generalized functions in the Zemanian space with the asymptotics of their fractional Hankel transform. Additionally, we establish both the initial and final value theorems for the fractional Hankel transform of distributions.

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arXiv2025

Chaos expansion solutions of a class of magnetic Schrödinger Wick-type stochastic equations on $\mathbb{R}^d$

We treat some classes of linear and semilinear stochastic partial differential equations of Schrödinger type on $\mathbb{R}^d$, involving a non-flat Laplacian, within the framework of white noise analysis, combined with Wiener-Itô chaos expansions and pseudodifferential operator methods. The initial data and potential term of the Schrödinger operator are assumed to be generalized stochastic processes that have spatial dependence. We prove that the equations under consideration have unique solutions in the appropriate (intersections of weighted) Sobolev-Kato-Kondratiev spaces.

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arXiv2024

Ultradifferentiable functions via the Laguerre operator

We define and characterize ultradifferentiable functions and their corresponding ultradistributions on $\RR^d_+$ using iterates of the Laguerre operator. The characterization is based on decay or growth conditions of the coefficients in their Laguerre series expansion. We apply our results to establish an isomorphism between subspaces of Pilipović spaces on $\RR^d$, and the spaces of ultradifferentiable functions on $\RR^d_+$.

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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

Ultradistributions on $\mathbb R_{+}^{n}$. Solvability and hypoellipticity through series expansions of ultradistributions

In the first part we analyze space $\mathcal G^*(\mathbb R^{n}_+)$ and its dual through Laguerre expansions when these spaces correspond to a general sequence $\{M_p\}_{p\in\mathbb N_0}$, where $^*$ is a common notation for the Beurling and Roumieu cases of spaces. In the second part we are solving equation of the form $Lu=f,\; L=\sum_{j=1}^ka_jA_j^{h_j}+cE^{d}_y+bP(x,D_x),$ where $f$ belongs to the tensor product of ultradistribution spaces over compact manifolds without boundaries as well as ultradistribution spaces on $\mathbb R^n_+$ and $\mathbb R^m$; $A_j, j=1,...,k$, $E_y$ and $P(x,D_x)$ are operators whose eigenfunctions form orthonormal basis of corresponding $L^2-$space. The sequence space representation of solutions enable us to study the solvability and the hypoellipticity in the specified spaces of ultradistributions.

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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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arXiv2024

Vector valued Hardy spaces related to analytic functions having distributional boundary values

The Hardy space $H^{p}$ of vector valued analytic functions in tube domains in $\mathbb{C}^{n}$ and with values in Banach space are defined. Vector valued analytic functions in tube domains in $\mathbb{C}^{n}$ with values in Hilbert space and which have vector valued tempered distributions as boundary value are proved to be in $H^{p}$ corresponding to Hilbert space if the boundary value is in $L^{p}$ with values in Hilbert space. A Poisson integral representation for such vector valued analytic functions is obtained.

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arXiv2024

On the iterates of the Laguerre operator

We use the iterates of the Laguerre operator to introduce Pilipović spaces on positive orthants. It is shown that such spaces coincide with $G-$type spaces $g_α^α(\mathbb{R}^d_+)$ and $G_α^α(\mathbb{R}^d_+)$, when $α> 1$, and $α\geq 1$, respectively. However, in contrast to $G$-type spaces, Pilipović spaces on positive orthants are nontrivial below the critical index $α= 1$. We also remark that there is a natural isomorphism between subspaces of Pilipović spaces on $\mathbb{R}^d$ consisting of even functions, and Pilipović spaces on positive orthants.

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arXiv2024

On the product of periodic distributions. Product in shift-invariant spaces

We connect through the Fourier transform shift-invariant Sobolev type spaces $V_s\subset H^s$, $s\in\mathbb R,$ and the spaces of periodic distributions and analyze the properties of elements in such spaces with respect to the product. If the series expansions of two periodic distributions have compatible coefficient estimates, then their product is a periodic tempered distribution. We connect product of tempered distributions with the product of shift-invariant elements of $V_s$. The idea for the analysis of products comes from the Hörmander's description of the Sobolev type wave front in connection with the product of distributions. Coefficient compatibility for the product of $f$ and $g$ in the case of "good" position of their Sobolev type wave fronts is proved in the 2-dimensional case. For larger dimension it is an open problem because of the difficulties on the description of the intersection of cones in dimension $d\geqslant3$.

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arXiv2024

Almost diagonalization of $Ψ$DO's over various generalized function spaces

Inductive and projective type sequence spaces of sub- and super-exponential growth, and the corresponding inductive and projective limits of modulation spaces are considered as a framework for almost diagonalization of pseudo-differential operators. Moreover, recent results of the first author and B. Prangoski related to the almost diagonalization of pseudo-differential operators in the context of Hörmander metrics are reviewed.

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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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arXiv2022

Besov regularity in non-linear generalized functions

We introduce and study new modules and spaces of generalized functions that are related to the classical Besov spaces. Various Schwartz distribution spaces are naturally embedded into our new generalized function spaces. We obtain precise criteria for detecting Besov regularity of distributions.

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