|M.Sc Student||Mark Kozdoba|
|Subject||Extension of Banach Space Valued Lipschitz Functions|
|Department||Department of Mathematics||Supervisor||Professor Emeritus Benyamini Yoav|
This work is devoted to a study of certain aspects of the following Extension Problem:
Give conditions on the metric spaces X , Y and the Banach space Z, where X is a subspace of Y, such that every Lipschitz function f from X to Z has a Lipschitz extension F from Y to Z (F is an extension of f iff the restriction of F to X equals f). Give upper bounds on the ratio between the Lipschitz norms of F and f.
In the first part of the work we present some known results. We start with two results by Johnson, Lindenstrauss and Schechtman, dealing with the cases where Y is a finite dimensional normed space or X is finite. In a recent work, Lee and Naor generalized and unified these results, by showing that if X is a metric space with a finite doubling constant then any Lipschitz function from X to any Banach space Z can be extended
to any metric space Y containing X with the loss in the Lipschitz constant controlled
by the doubling constant of X only. The doubling constant of a metric space is a parameter analogous to the dimension in the normed spaces category. This result is also presented in the first part of the work. In addition, we give an alternative proof
to one of the main lemmas.
For a metric space X, we define e(X) to be the infimum over all constants C>0 with the property that for every Banach space Z, every metric space Y containing X and every Lipschitz function f from X to Z, there exists an extension F from Y to Z such that the Lipschitz constant of F is bounded by C times the Lipschitz constant of f.
The result of Lee and Naor implies that e(X) is finite if X has a finite doubling
constant. There also are other examples of spaces with finite e(X). The second
part of this work is devoted to a study of the parameter e(X). We show that it can be
obtained as an extension constant of a specific map between a pair of specific “universal" spaces associated to X, the Free Banach space generated by X, and the injective envelope of X. Then we proceed to derive various useful properties of e(X) as a function of the space X and we apply those results to derive estimates on e(X) in some simple cases.