On Whitney-type extension theorems on Banach spaces for C1,ω,C1,+, C1,+ loc , and C1,+ B-smooth functions

Our paper is a complement to a recent article by D. Azagra and C. Mudarra (2021, [2]). We show how older results on semiconvex functions with modulus omega easily imply extension theorems for C1,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 C1,omega-smooth functions from open quasiconvex sets. They proved also an extension theorem for C1,+ B-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 C1,+ and C1,+ loc-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. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

Our paper complements a recent article by D. Azagra and C. Mudarra (2021, [2]). We demonstrate how previous results on semiconvex functions with modulus omega can easily lead to extension theorems for C1,omega-smooth functions on super-reflexive Banach spaces, which are variations of Azagra and Mudarra's theorems. We also present some new interesting consequences that were not mentioned in their article, particularly extensions of C1,omega-smooth functions from open quasiconvex sets. Our paper contributes further by presenting our version of their extension theorem for C1,+ B-smooth functions (i.e., functions with uniformly continuous derivative on each bounded set) on Hilbert spaces, as well as new extension results for C1,+ and C1,+ loc-smooth functions (i.e., functions with uniformly and locally uniformly continuous derivative) on arbitrary super-reflexive Banach spaces. Some of our proofs are inspired by the ideas of Azagra and Mudarra's article, but are formally independent of their work.