# The Kurdyka-\L{}ojasiewicz inequality as regularity condition

## TL;DR

This paper demonstrates that the Kurdyka-ojasiewicz inequality can serve as a regularity condition in Tikhonov regularization, linking it to existing smoothness and rate conditions in Banach spaces.

## Contribution

It establishes the equivalence between the KL inequality and various known regularity conditions, providing a unified framework for convergence analysis.

## Key findings

- KL inequality is equivalent to known regularity conditions
- Theoretical link between KL inequality and convergence rates
- Illustrative examples with source conditions and stability estimates

## Abstract

We show that a Kurdyka-\L{}ojasiewicz (KL) inequality can be used as regularity condition for Tikhonov regularization with linear operators in Banach spaces. In fact, we prove the equivalence of a KL inequality and various known regularity conditions (variational inequality, rate conditions, and others) that are utilized for postulating smoothness conditions to obtain convergence rates. Case examples of rate estimates for Tikhonov regularization with source conditions or with conditional stability estimate illustrate the theoretical result.