Lipschitz and Fourier type conditions with moduli of continuity in rank 1 symmetric spaces

TL;DR

This paper investigates Lipschitz and Fourier transform conditions for functions on rank 1 symmetric spaces, establishing necessary and sufficient criteria involving moduli of continuity.

Contribution

It provides new necessary and sufficient conditions linking Lipschitz-type integrals and Fourier bounds in rank 1 symmetric spaces with growth depending on higher-order moduli of continuity.

Findings

Established equivalence between Lipschitz conditions and Fourier transform bounds.

Extended classical results to rank 1 symmetric spaces with higher-order moduli.

Provided a comprehensive framework for analyzing function regularity in geometric analysis.

Abstract

Sufficient and necessary results have been proven on Lipschitz type integral conditions and bounds of its Fourier transform for an $L^2$ function, in the setting of Riemannian symmetric spaces of rank $1$ whose growth depends on a $k$th-order modulus of continuity.