On a unique two-dimensional integral operator homogeneous with respect to all orientation preserving linear transformations
Zhirayr Avetisyan, Alexey Karapetyants, Adolf Mirotin
TL;DR
The paper identifies and analyzes the unique $2$-D homogeneous, antisymmetric integral operator ${\mathbf K}$ associated with orientation-preserving linear transformations, establishing a polar representation ${\mathcal K}$ and showing ${\mathbf K}=S^{-1}{\mathcal K}S$ via a coordinate intertwiner $S$. It clarifies the operator’s geometric origin from homogeneous spaces and connects it to Hilbert and Radon transforms, while providing rigorous boundedness results on projective tensor products of classical spaces. The key contributions include explicit norm estimates in terms of $v_p(x_1,x_2)=|x_1|^{-1/q}|x_2|^{-1/p}$ and the Riesz constants, and a comprehensive framework for mapping properties on Cartesian and polar tensor-product domains. This work advances the theory of homogeneous integral operators in dimension two and offers precise criteria for their action between Banach lattices and tensor-product spaces, with implications for harmonic analysis and related transforms.
Abstract
In this paper, we consider a two-dimensional operator with an antisymmetric integral kernel, recently introduced by Z. Avetisyan and A. Karapetyants in connection to the study of general homogeneous operators. This is the unique two-dimensional operator that has an antisymmetric kernel homogeneous with respect to all orientation-preserving linear transformations of the plane. It is shown that the operator under consideration interacts naturally, both in Cartesian and polar coordinates, with projective tensor products of some classical functional spaces, such as Lebesgue, Hardy, and Hölder spaces; conditions for their boundedness as operators acting from these spaces to Banach lattices of measurable functions and estimates of their norms are given.