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

This article is cited in 1 scientific paper (total in 1 paper)

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)
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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}


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    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  mathnet  crossref  crossref  isi  elib
  • Mathematical Physics and Computer Simulation
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