Homogeneous spaces adapted to singular integral operators involving rotations

TL;DR

This paper develops new homogeneous spaces incorporating rotations to analyze singular integral operators, proving boundedness results on Lie groups like the Heisenberg group, with applications to hydrodynamics.

Contribution

It introduces novel spaces of homogeneous type adapted to rotational symmetries, enabling Calderón-Zygmund analysis for operators involving rotations on Lie groups.

Findings

Proved weak-type (1,1) bounds for operators on homogeneous Lie groups.

Established L^p estimates for operators involving rotations.

Applied results to hydrodynamical problems with rotational components.

Abstract

Calder\'on-Zygmund decompositions of functions have been used to prove weak-type (1,1) boundedness of singular integral operators. In many examples, the decomposition is done with respect to a family of balls that corresponds to some family of dilations. We study singular integral operators $T$ that require more particular families of balls, providing new spaces of homogeneous type. Rotations play a decisive role in the construction of these balls. Boundedness of $T$ can then be shown via Calder\'on-Zygmund decompositions with respect to this space of homogeneous type. We prove weak-type (1,1) and $\LP^p$ estimates for operators $T$ acting on $\LP^p(G)$, where $G$ is a homogeneous Lie group. Our results apply to the setting where the underlying group is the Heisenberg group and the rotations are symplectic automorphisms. They also apply to operators that arise from some hydrodynamical problem involving rotations.