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Đorđe Vučković

Publications and source records attributed to Đorđe Vučković.

10 recordsLinked to original sources
arXiv2026

Continuity properties of the Laguerre operator and its propagator

We study the well-posedness of a Cauchy problem associated with the general form of the Laguerre operator and relate it to the corresponding global problem for the harmonic oscillator. To this end, we carry out a detailed analysis of the continuity properties of the associated propagator. Furthermore, we establish connections between several integral transforms, including the fractional Fourier transform and the fractional Hankel transform. Our results highlight the role of Pilipović spaces on positive orthants when studying problems involving the Laguerre operator.

math.AP↗
arXiv2026

An interpolation problem for extended Gevrey regularity

We study an interpolation problem in extended Gevrey classes defined by the sequences $M_n^{τ,σ}=n^{τn^σ}$, $n\in\mathbb N$, $τ>0$, $σ>1$. We show that, under a suitable growth condition on a divergent sequence of derivative orders, estimates of extended Gevrey type imposed only along this sequence imply corresponding estimates for all derivative orders. In particular, we explicitly quantify the change of the parameters in the resulting estimates. To this end, we first establish an equivalence between two definitions of the extended Gevrey classes, one involving the supergeometric factor $h^{n^σ}$ and another in which this factor is omitted. We then establish a corresponding interpolation principle for extended Gelfand--Shilov spaces, including a symmetric characterization and a formulation in terms of the associated function of the defining sequence.

math.FA↗
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 $\RR^d_+$.

math.FA↗
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.

math.FA↗
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 $\exp(\exp(l|\cdot|^q))$, $l,q>0$.

math.FA↗
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.

math.FA↗
arXiv2017

Ultradistributional boundary values of harmonic functions on the sphere

We present a theory of ultradistributional boundary values for harmonic functions defined on the Euclidean unit ball. We also give a characterization of ultradifferentiable functions and ultradistributions on the sphere in terms of their spherical harmonic expansions. To this end, we obtain explicit estimates for partial derivatives of spherical harmonics, which are of independent interest and refine earlier estimates by Calderon and Zygmund. We apply our results to characterize the support of ultradistributions on the sphere via Abel summability of their spherical harmonic expansions.

math.FA↗
arXiv2017

Rotation invariant ultradistributions

We prove that an ultradistribution is rotation invariant if and only if it coincides with its spherical mean. For it, we study the problem of spherical representations of ultradistributions on $\mathbb{R}^{n}$. Our results apply to both the quasianalytic and the non-quasianalytic case.

math.FA↗
arXiv2016

Eigenfunction expansions of ultradifferentiable functions and ultradistributions in $\mathbb R^n$

We obtain a characterization of ${\mathcal S}^{\{M_p\}}_{\{M_p\}}(\mathbb R^n)$ and $\mathcal {S}^{(M_p)}_{(M_p)}(\mathbb {R}^n)$, the general Gelfand-Shilov spaces of ultradifferentiable functions of Roumieu and Beurling type, in terms of decay estimates for the Fourier coefficients of their elements with respect to eigenfunction expansions associated to normal globally elliptic differential operators of Shubin type. Moreover, we show that the eigenfunctions of such operators are absolute Schauder bases for these spaces of ultradifferentiable functions. Our characterization extends earlier results by Gramchev et al. (Proc. Amer. Math. Soc. 139 (2011), 4361-4368) for Gevrey weight sequences. It also generalizes to $\mathbb{R}^{n}$ recent results by Dasgupta and Ruzhansky which were obtained in the setting of compact manifolds.

math.AP↗