Self-improvement phenomena are important both for theoretical and practical reasons. In analysis of metric spaces, Keith and Zhong on the one hand and Lewis on the other proved foundational self-improvement results. Their results are hard, and the realization of this project has been that all of these self-improvement results can be cast in the same framework as a task of constructing a “sufficiently good curve”, which is obtained via an iteration based on filling and choosing an appropriate level. The new insights gained allow for a new weighted result with Vähäkangas and Lehrbäck, which was not known before. It further gives a way of fairly directly obtaining the full strength of Keith and Zhong as well as Lewis, with essentially sharp bounds on the self-improvement.
Papers:
- Self-improvement of weighted pointwise inequalities on open sets, joint with Juha Lehrbäck and Antti V. Vähäkangas, submitted 2019
- Self-improvement of pointwise Hardy inequality, joint with Antti V. Vähäkangas, Trans. Amer. Math. Soc.(2019)
- Alternative proof of Keith-Zhong self-improvement and Conenctivity. Ann. Acad. Sci. Fenn.
- An alternative version and perspective to Keith-Zhong was also presented here.
Quasiworld Network - Sylvester E-B 