Reference

Yes, there are variations of Tietze's extension theorem that apply to Hölder continuous functions. The Tietze extension theorem generally states that any real-valued, continuous function defined on a closed subset of a normal topological space can be extended to a continuous function on the whole space. Variations of this theorem have been explored for functions that are not just continuous, but also satisfy certain conditions such as Hölder continuity.

The search results indicate that there are indeed extensions of the Tietze's theorem for functions with moduli of uniform continuity, which includes Hölder continuous functions. For instance, a paper by Shafer discusses the reverse mathematics of the Tietze extension theorem for functions with moduli of uniform continuity, proving that several versions of the theorem are equivalent to WKL0 over RCA0 [Source: https://eprints.whiterose.ac.uk/111928/1/ShaferRMTietzeExtension.pdf].

Another example is the work on harmonious extensions in metrically convex compact metric spaces, which provides a class of optimally stable solutions to Tietze's extension problem [Source: https://epubs.siam.org/doi/abs/10.1137/S0036141095294067]. This suggests that the theorem can be adapted for specific types of spaces and functions, including those with Hölder continuity.

In summary, while the classical Tietze extension theorem deals with continuous functions, there are variations and generalizations that extend its applicability to include Hölder continuous functions, providing a broader framework for function extension in various mathematical contexts.