# Calder\'on-type inequalities for affine frames

## TL;DR

This paper establishes precise bounds for Calderón-type sums in affine frames on LCA groups, utilizing metric space analysis and lattice point counting, with applications to automorphisms and Gabor systems.

## Contribution

It introduces sharp bounds for Calderón sums in affine frames on LCA groups, extending to automorphisms and Gabor systems, using novel metric space analysis techniques.

## Key findings

- Derived sharp bounds for Calderón sums in affine frames.
- Extended results to automorphisms and Gabor systems.
- Utilized lattice point counting in metric spaces.

## Abstract

We prove sharp upper and lower bounds for generalized Calder\'on's sums associated to frames on LCA groups generated by affine actions of cocompact subgroup translations and general measurable families of automorphisms. The proof makes use of techniques of analysis on metric spaces, and relies on a counting estimate of lattice points inside metric balls. We will deduce as special cases Calder\'on-type inequalities for families of expanding automorphisms as well as for LCA-Gabor systems.