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

Let C� be a �-dimensional complex coordinate space and let Rbe a set of real numbers. The Heisenberg group is a set H� = C� × R with the binary operation(�, 𝑎)(�, 𝑏) = (� + �, + + 2Im(� · �)), (�, 𝑎), (�, 𝑏) ∈ H�.The group under consideration is endowed with a family of dilationsδ�(�, 𝑎) = (��, �2𝑎), ∈ R+, (�, 𝑎) ∈ H�,and is equipped with the Koranyi norm �‖(�, 𝑎)‖ = (︀|�|4 + 𝑎2)︀ 1, (�, 𝑎) ∈ H.This norm allows us to define the notion of the unit ball on the Heisenberg groupS� = {� ∈ H� : ‖�‖ = 1}.The transformation of Cartesian coordinates on the Heisenberg group� ∈ H� ∖ {(0, 0)} to spherical coordinates (�, �) ∈ R+ × S� is defined by‖ ‖� = �, = δ-1 (�).‖�‖× →The function : H� H� C is said to be homogeneous of degree if it satisfies the condition of homogeneity�∀γ ∈ R+, ∀�, ∈ H� : �(δγ(�), δγ(�)) = γ �(�, �).This paper is concerned with the study of linear integral operators on the Heisenberg group of the form∫︁(𝐾� )(�) =H��(�, �) (�) 𝑑�,� ⊂ ℳ∞- -ℳwhere function is an element of the special Banach space �(H�) of homogeneous ( 2� 2) degree functions. It is claimed that operator under consideration is bounded in the space 𝐿�(H�), where 1 function show_eabstract() { $('#eabstract1').hide(); $('#eabstract2').show(); $('#eabstract_expand').hide(); }