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Stevan Pilipovic

Publications and source records attributed to Stevan Pilipovic.

At least 19 recordsLinked to original sources
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

math.FA↗
arXiv2024

Topology and regularity for generalized ultradistribution algebras

Compiling essential results for non-quasianalytic ultradistribution spaces and Colombeau versions of generalized ultradistribution algebras, we analyze strong $B$- and strong $R$-association of a generalized ultradistribution $[(f_\varepsilon)]$. The strong association of $[(f_\varepsilon)]$ to a Komatsu-type ultradistribution $T$, with additional assumption on regularity of $[(f_\varepsilon)]$ of Beurling, respectively, Roumieu type, implies that $T$ is an ultradifferentiable function of Beurling, Roumieu type, respectively. We demonstrate that, under suitable conditions on regularity, a weakly negligible net $(g_\varepsilon)_{\varepsilon\in(0,1)}$ (meaning that the net of complex numbers $(\int g_\varepsilon\phi dx)_{\varepsilon\in(0,1)}$ is Beurling, respectively, Roumieu negligible for every ultradifferentiable function $\phi$ in the corresponding test space), is a negligible net in the sense of generalized ultradistributions. Furthermore, we prove that a translation invariant generalized ultradistribution $g$ is equal to a generalized constant in both types of generalized ultradistribution algebras.

math.FA↗
arXiv2023

Stochastic evolution equations with Wick-analytic nonlinearities

We study nonlinear stochastic partial differential equations with Wick-analytic type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fisher--KPP equations, stochastic Allen--Cahn, stochastic Newell--Whitehead--Segel, and stochastic Fujita--Gelfand equations. By implementing the theory of $C_0-$semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove existence and uniqueness of solutions for this class of stochastic partial differential equations.

math.PR↗
arXiv2023

Stochastic evolution equations with Wick-polynomial nonlinearities

We study nonlinear parabolic stochastic partial differential equations with Wick-power and Wick-polynomial type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fujita equation, the stochastic Fisher-KPP equation and the stochastic FitzHugh-Nagumo equation among many others. By implementing the theory of $C_0-$semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove existence and uniqueness of solutions for this class of SPDEs. In particular, we also treat the linear nonautonomous case and provide several applications featured as stochastic reaction-diffusion equations that arise in biology, medicine and physics.

math.PR↗
arXiv2023

Stochastic evolution equations with multiplicative noise

We study parabolic stochastic partial differential equations (SPDEs), driven by two types of operators: one linear closed operator generating a $C_0-$semigroup and one linear bounded operator with Wick-type multiplication, all of them set in the infinite dimensional space framework of white noise analysis. We prove existence and uniqueness of solutions for this class of SPDEs. In particular, we also treat the stationary case when the time-derivative is equal to zero.

math.PR↗
arXiv2023

Generalized exponentially bounded integrated semigroups

The main subject of this paper is the analysis of sequences of exponentially bounded integrated semigroups which are related to Cauchy problems \begin{equation}\label{jed} \frac{\partial}{\partial t}u(t,x)-a(D)u(t,x)=f(t,x), \quad u(0,x)=u_0(x), \quad t\geq 0, \ x\in \mathbb R^d, \end{equation} with a distributional initial data $u_0$ and a distributional right hand side $f$ through a sequence of equations with regularized $u_0$ and $f$ and a sequence of (pseudo) differential operators $a_n(D)$ instead of $a(D)$. Comparison of sequences of infinitesimal generators and the determination of corresponding sequences of integrated semigroups are the main subject of the paper. For this purpose, we introduce association, the relation of equivalence for infinitesimal generators on one side and the corresponding relations of equivalence of integrated semigroups on another side. The order of involved assumptions on generators essentially characterize the mutual dependence of sequences of infinitesimal generators and the corresponding sequences of integrated semigroups.

math.FA↗
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 $\mathbb{R}^{2d}$.

math.FA↗
arXiv2018

New classes of weighted Hölder-Zygmund spaces and the wavelet transform

We provide a new and elementary proof of the continuity theorem for the wavelet and left-inverse wavelet transforms on the spaces $ \mathcal{S}_0(\mathbb{R}^n) $ and $ \mathcal{S}(\mathbb{H}^{n+1})$. We then introduce and study a new class of weighted Hölder-Zygmund spaces, where the weights are regularly varying functions. The analysis of these spaces is carried out via the wavelet transform and generalized Littlewood-Paley pairs.

math.FA↗
arXiv2017

On quasianalytic classes of Gelfand-Shilov type. Parametrix and convolution

We develop a convolution theory for quasianalytic ultradistributions of Gelfand-Shilov type. We also construct a special class of ultrapolynomials, and use it as a base for the parametrix method in the study of new topological and structural properties of several quasianalytic spaces of functions and ultradistributions. In particular, our results apply to Fourier hyperfunctions and Fourier ultra-hyperfunctions.

math.FA↗
arXiv2017

Directional short-time Fourier transform and directional regularity

We give some new results related to the directional short-time Fourier transform (DSTFT) and extend them on the spaces $\mathcal K_{1}(\mathbb R^{n})$ and $\mathcal K_{1}({\mathbb R})\widehat{\otimes}\mathcal U(\mathbb C^n)$ and their duals. Then, we define multi-directional STFT and, for tempered distributions, directional regular sets and their complements, directional wave fronts. Different windows with mild conditions on their support show the invariance of these notions related to window functions. Smoothness of $f$ follows from the assumptions of the directional regularity in any direction.

math.FA↗
arXiv2017

Wave-front sets related to quasi-analytic Gevrey sequences

Quasi-analytic wave-front sets of distributions which correspond to the Gevrey sequence $p!^s$, $s\in[1/2,1)$ are defined and investigated. The propagation of singularities are deduced by considering sequences of Gaussian windowed short-time Fourier transforms of distributions which are modifications of the original distributions by suitable restriction-extension techniques. Basic micro-local properties of the new wave-fronts are thereafter established.

math.AP↗
arXiv2016

Anisotropic Shubin operators and eigenfunctions expansions in Gelfand-Shilov spaces

We derive new results on the characterization of Gelfand--Shilov spaces $\mathcal{S}^μ_ν(\R^n)$, $μ,ν>0$, $μ+ν\geq 1$ by Gevrey estimates of the $L^2$ norms of iterates of $(m,k)$ anisotropic globally elliptic Shubin (or $Γ$) type operators, $(-Δ)^{m/2} +| x |^k$ with $m,k\in 2\N$ being a model operator, and on the decay of the Fourier coefficients in the related eigenfunction expansions. Similar results are obtained for the spaces $Σ^μ_ν(\R^n)$, $μ,ν>0$, $μ+ν> 1$, cf. \eqref{GSdef}. In contrast to the symmetric case $μ= ν$ and $k=m$ (classical Shubin operators) we encounter resonance type phenomena involving the ratio $κ:=μ/ν$; namely we obtain a characterization of $\mathcal{S}^μ_ν(\R^n)$ and $Σ^μ_ν(\R^n)$ in the case $μ=kt/(k+m), ν= mt/(k+m), t \geq 1$, that is, when $κ=k/m \in \Q$.

math.FA↗
arXiv2016

H-distribution via Sobolev spaces

H-distributions associated to weakly convergent sequences in Sobolev spaces are determined. It is shown that a weakly convergent sequence $(u_n)$ in $W^{-k,p}( \R^d)$ has the property that $θu_n$ converges strongly in $W^{-k,p}(\R^d)$ for every $θ\in\mathcal S(\R^d)$ if and only if all H-distributions related to this sequence are equal to zero. Results are applied on a weakly convergent sequence of solutions to a family of linear first order PDEs.

math.AP↗
arXiv2016

Convolution of ultradistributions and ultradistribution spaces associated to translation-invariant Banach spaces

We introduce and study a number of new spaces of ultradifferentiable functions and ultradistributions and we apply our results to the study of the convolution of ultradistributions. The spaces of convolutors $\mathcal{O}'^{\ast}_{C}(\mathbb{R}^{d})$ for tempered ultradistributions are analyzed via the duality with respect to the test function spaces $\mathcal{O}^{\ast}_{C}(\mathbb{R}^{d})$, introduced in this article. We also study ultradistribution spaces associated to translation-invariant Banach spaces of tempered ultradistributions and use their properties to provide a full characterization of the general convolution of Roumieu ultradistributions via the space of integrable ultradistributions. We show that the convolution of two Roumieu ultradistributions $T,S\in \DD'^{\{M_p\}}\left(\RR^d\right)$ exists if and only if $\left(φ*\check{S}\right)T\in\DD'^{\{M_p\}}_{L^1}\left(\RR^d\right)$ for every $φ\in\DD^{\{M_p\}}\left(\RR^d\right)$.

math.FA↗
arXiv2015

The wavelet transforms in Gelfand-Shilov spaces

We describe local and global properties of wavelet transforms of ultradifferentiable functions. The results are given in the form of continuity properties of the wavelet transform on Gelfand-Shilov type spaces and their duals. In particular, we introduce a new family of highly time-scale localized spaces on the upper half-space. We study the wavelet synthesis operator (the left-inverse of the wavelet transform) and obtain the resolution of identity (Calderón reproducing formula) in the context of ultradistributions.

math.FA↗