RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mathematical Physics and Computer Simulation: Year: Volume: Issue: Page: Find

 Mathematical Physics and Computer Simulation, 2018, Volume 21, Issue 3, Pages 5–18 (Mi vvgum233)

Mathematics and mechanics

The invertibility of integral operators with homogeneous kernels of compact type on the heisenberg group

V. V. Denisenkoa, V. M. Deundyakbc

a Southern Federal University
b Southern Federal University, Rostov-on-Don
c Specvuzavtomatika Research Institute

Abstract: Let $\mathbb{C}^n$ be a $n$–dimensional complex coordinate space and let $\mathbb{R}$ be a set of real numbers. The Heisenberg group is a set $\mathbb{H}_{n} = \mathbb{C}^n \times \mathbb{R}$ with the binary operation
\begin{equation*} (z,a) (w, b) = (z + w, a + b + 2 \mathrm{Im}(z \cdot w)), \quad (z,a), (w,b) \in \mathbb{H}_n. \end{equation*}
The group under consideration is endowed with a family of dilations
\begin{equation*} \delta_r(z,a) = (r z, r^2 a), \quad r \in \mathbb{R}_{+}, \quad (z,a) \in \mathbb{H}_n, \end{equation*}
and is equipped with the Koranyi norm
\begin{equation*} łVert (z,a) \rVert = (\lvert z \rvert ^4 + a^2 )^{\frac{1}{4}}, \quad (z,a) \in \mathbb{H}_n. \end{equation*}
This norm allows us to define the notion of the unit ball on the Heisenberg group
\begin{equation*} \mathbb{S}_{n} = \{ x \in \mathbb{H}_{n} : \: łVert x \rVert = 1 \}. \end{equation*}
The transformation of Cartesian coordinates on the Heisenberg group $x \in \mathbb{H}_n \setminus \{ (\mathbf{0}, 0) \}$ to spherical coordinates $(r,s) \in \mathbb{R}_+ \times \mathbb{S}_n$ is defined by
\begin{equation*} r = łVert x \rVert, \quad s = \delta_{łVert x \rVert}^{-1}(x). \end{equation*}

The function $k: \mathbb{H}_{n} \times \mathbb{H}_{n} \to \mathbb{C}$ is said to be homogeneous of degree $m$ if it satisfies the condition of homogeneity
\begin{equation*} \forall \gamma \in \mathbb{R}_{+}, \quad \forall x,y \in \mathbb{H}_{n}: \quad k(\delta_{\gamma} (x), \delta_{\gamma} (y)) = \gamma^{m} k(x,y). \end{equation*}
This paper is concerned with the study of linear integral operators on the Heisenberg group of the form
\begin{equation*} (K_{k}   f)(x) = \int\limits_{\mathbb{H}_n} k(x,y)   f(y)   dy, \end{equation*}
where function $k$ is an element of the special Banach space $\mathcal{M}_{p}(\mathbb{H}_n)$ of homogeneous $(-2n-2)$ degree functions. It is claimed that operator under consideration is bounded in the space $L_p(\mathbb{H}_n)$, where $1 < p < \infty$.
A new class $\mathcal{C}_{p}(\mathbb{H}_n) \subset \mathcal{M}_{p}(\mathbb{H}_n)$ of homogeneous kernels of compact type is introduced. The main object of the research is the unitary Banach algebra $\mathfrak{V}_{p}^+(\mathbb{H}_n)$ generated by integral operators with $\mathcal{C}_{p}(\mathbb{H}_n)$ kernels. It should be pointed out that spherical coordinate system on the Heisenberg group plays a significant role in construction of the $\mathcal{C}_{p}(\mathbb{H}_n)$ class.
The convolutional representation of the unitary Banach algebra $\mathfrak{V}_{p}^+(\mathbb{H}_n)$ is constructed using the technique of tensor products. This representation makes it possible to define the symbol for integral operators in $\mathfrak{V}_{p}^+(\mathbb{H}_n)$ algebra and formulate the necessary and sufficient conditions for invertibility of these operators in terms of their symbol.

Keywords: Heisenberg group, linear integral operators, operators with homogeneous kernels, convolutional representation, symbolic calculus, invertibility of operators.

DOI: https://doi.org/10.15688/mpcm.jvolsu.2018.3.1

Full text: PDF file (389 kB)
References: PDF file   HTML file

UDC: 517.983
BBK: 22.162

Citation: V. V. Denisenko, V. M. Deundyak, “The invertibility of integral operators with homogeneous kernels of compact type on the heisenberg group”, Mathematical Physics and Computer Simulation, 21:3 (2018), 5–18

Citation in format AMSBIB
\Bibitem{DenDeu18} \by V.~V.~Denisenko, V.~M.~Deundyak \paper The invertibility of integral operators with homogeneous kernels of compact type on the heisenberg group \jour Mathematical Physics and Computer Simulation \yr 2018 \vol 21 \issue 3 \pages 5--18 \mathnet{http://mi.mathnet.ru/vvgum233} \crossref{https://doi.org/10.15688/mpcm.jvolsu.2018.3.1} 

• http://mi.mathnet.ru/eng/vvgum233
• http://mi.mathnet.ru/eng/vvgum/v21/i3/p5

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. V. Denisenko, V. M. Deundyak, “Fredholm Property of Integral Operators with Homogeneous Kernels of Compact Type in the $L_2$ Space on the Heisenberg Group”, Proc. Steklov Inst. Math., 308 (2020), 155–167
•  Number of views: This page: 128 Full text: 25 References: 5