Abstract
In Chapter 4, we proved the Poincaré inequality in ℝn by using the fact that points in ℝn can be joined by a thick “pencil” of curves. In the previous chapter, we defined the Loewner function of a metric space that detects quantitatively whether or not the space contains rectifiable curves. In this chapter, we will see that these two concepts, the Poincaré inequality and the Loewner condition, are related.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media New York
About this chapter
Cite this chapter
Heinonen, J. (2001). Loewner Spaces and Poincaré Inequalities. In: Lectures on Analysis on Metric Spaces. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0131-8_9
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0131-8_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6525-2
Online ISBN: 978-1-4613-0131-8
eBook Packages: Springer Book Archive