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

V. V. Denisenko ; V. M. Deundyak

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 21 (2018) no. 3, pp. 5-18

Résumé

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*} \lVert (z,a) \rVert = \left(\lvert z \rvert ^4 + a^2 \right)^{\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} = \left\{ x \in \mathbb{H}_{n} : \: \lVert x \rVert = 1 \right\}. \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 = \lVert x \rVert, \quad s = \delta_{\lVert 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.

Détail

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