Isometric Extensions
Let A⊂B, and let Y be a Banach function space on B. Two common
interpolation problems are:
(i) Given f:A→\C, find g∈Y so that g|A=f and g has the smallest possible norm.
(ii) Let X be a Banach function space on A. Determine if there is a constant M, and if so what the best value is, so that for every f:A→\C in X there exists g∈Y so that g|A=f and ‖g‖Y≤M‖f‖X.
It is often the case in Problem (i) that the optimal solution has extra regularity properties. We consider an analogous question for Problem (ii).
Suppose B is a pseudoconvex set in \Cn, and A is an analytic subvariety. Let X and Y be the bounded holomorphic functions on A and B respectively. If the constant M can be chosen to be 1, what does this say about the relationship between A and B?
If B is nice, eg. the ball, then A is rigidly constrained: it must be a holomorphic retract. But {\em every } A can occur for some B.
We shall discuss both the nice and non-restricted cases. The talk is base on joint work with J. Agler and \L. Kosi\'nski.