Local behavior of two-sided convolution operators with singular kernels on the Heisenberg group
The present paper gives an extended version of some results announced by the author in [Russ. Acad. Sci., Dokl. Math. 46, 12-16 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 325, 20-23 (1992; Zbl 0801.47037)] and [Russ. Acad. Sci., Dokl. Math. 50, 92-97 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 337, 439-441 (1994)]. It combines PDO calculus techniques, on the one side, and localization techniques of I. B. Simonenko, on the other, to investigate operators which arise in noncommutative harmonic analysis. Two-sided convolution operators of the following type on the Heisenberg group \(\mathbb{H}^n\) are considered: \[ K = (2 \pi)^{-N} \int_{\mathbb{H}^n} \int_{\mathbb{H}^n} k(g_1, g_2) \pi_l (g_1) \pi_r (g_2) dg_1dg _2, \] where \(\pi_l\) and \(\pi_r\) are the left (right) shift operators on \(\mathbb{H}^n \), \(N = 2n + 1 = \dim \mathbb{H}^n\); the kernels \(k\) have Fourier images \(\widehat k\), non-isotropically homogeneous of degree zero with respect to dilations of \(\mathbb{H}^n \times \mathbb{H}^n\) and continuous on \(\Omega^{2n} \times \Omega^{2n}\) (here \(\Omega^{2n} = \{(t,z) \in \mathbb{R} \times \mathbb{C}^n : t^2 + |z |^4 = 1\}\) is the non-isotropical unit sphere, in exponential coordinates of the group \(\mathbb{H}^n)\). Such operators are bounded as mappings \(L^2_{\text{com }} (\mathbb{H}^n) \to L^1_{\text{loc}} (\mathbb{H}^n)\). The question of the Noetherity of such operators can be reduced to the question of the Noetherity of local representatives of these operators at the points of \(\mathbb{H}^n\). The main result is the following: At the points of the center of the group \(\mathbb{H}^n\) the operator \(K\) can not be simplified locally, at the other points these operators are locally equivalent to the sum of three operators: a left convolution on \(\mathbb{H}^n\), a right convolution on \(\mathbb{H}^n\), and a usual (Euclidean) convolution on \(\mathbb{R}^N\) \((= \mathbb{H}^n\) as a manifold). This result clarifies the structure of two-sided convolution operators with kernels from the above-mentioned class and reduces the investigation of them to well-investigated classes of one-sided convolution operators (see the corresponding bibliography in the paper).
scientific article; zbMATH DE number 830706