On Whitney-type extension theorems on Banach spaces for $C^{1,\omega}$, $C^{1,+}$, $C_{\mathrm{loc}}^{1,+}$, and $C_{\mathrm B}^{1,+}$-smooth functions

Abstract

Our paper is a complement to a recent article by D. Azagra and C. Mudarra (2021). We show how older results on semiconvex functions with modulus $\omega$ easily imply extension theorems for $C^{1,\omega}$-smooth functions on super-reflexive Banach spaces which are versions of some theorems of Azagra and Mudarra. We present also some new interesting consequences which are not mentioned in their article, in particular extensions of $C^{1,\omega}$-smooth functions from open quasiconvex sets. They proved also an extension theorem for $C_{\mathrm B}^{1,+}$-smooth functions (i.e., functions with uniformly continuous derivative on each bounded set) on Hilbert spaces. Our version of this theorem and new extension results for $C^{1,+}$ and $C_{\mathrm{loc}}^{1,+}$-smooth functions (i.e., functions with uniformly, resp. locally uniformly continuous derivative), all of which are proved on arbitrary super-reflexive Banach spaces, are further main contributions of our paper. Some of our proofs use main ideas of the article by D. Azagra and C. Mudarra, but all are formally completely independent on their article.