In this article on variational regularization for ill-posed nonlinear problems, we are once again discussing the consequences of an oversmoothing penalty term. This means in our model that the searched-for solution of the considered nonlinear operator equation does not belong to the domain of definition of the penalty functional. In the past years, such variational regularization has been investigated comprehensively in Hilbert scales, but rarely in a Banach space setting. Our present results try to establish a theoretical justification of oversmoothing regularization in Banach scales. This new study includes convergence rates results for a priori and a posteriori choices of the regularization parameter, both for H\"older-type smoothness and low order-type smoothness. An illustrative example is intended to indicate the specificity of occurring non-reflexive Banach spaces.

翻译：本文针对不适定非线性问题的变分正则化，再次讨论了过平滑惩罚项带来的影响。这意味着在我们的模型中，所考虑的非线性算子方程的解不属于惩罚泛函的定义域。过去几年中，这种变分正则化已在Hilbert尺度下得到全面研究，但在Banach空间框架下的研究却鲜有涉及。我们当前的研究试图为Banach尺度下的过平滑正则化建立理论依据。这项新研究包括正则化参数先验和后验选择的收敛速率结果，涵盖了Hölder型光滑性和低阶型光滑性。最后通过一个示例说明所涉及的非自反Banach空间的特殊性。