The Kurdyka-L{}ojasiewicz inequality as regularity condition

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.