Bj\"ork-Sj\"olin condition for strongly singular convolution operators on graded Lie groups

TL;DR

This paper extends the Bj"ork-Sj"olin theory for strongly singular convolution operators from Euclidean spaces to arbitrary graded Lie groups, using oscillating Hörmander conditions and Fourier transform decay measured via Rockland operators.

Contribution

It generalizes the classical Bj"ork-Sj"olin criteria to graded Lie groups, providing new conditions based on group Fourier analysis and Rockland operators.

Findings

Criteria for boundedness of convolution operators on graded Lie groups.

Reproduction of classical Euclidean results as a special case.

Framework for analyzing singular integrals on non-commutative groups.

Abstract

In this work we extend the $L^1$-Bj\"ork-Sj\"olin theory of strongly singular convolution operators to arbitrary graded Lie groups. Our criteria are presented in terms of the oscillating H\"ormander condition due to Bj\"ork and Sj\"olin of the kernel of the operator, and the decay of its group Fourier transform is measured in terms of the infinitesimal representation of an arbitrary Rockland operator. The historical result by Bj\"ork and Sj\"olin is re-obtained in the case of the Euclidean space.