Pavel Shvartsman

The Whitney problem of existence of a linear extension operator

We prove that a linear bounded extension operator exists for the trace of C1·ω (Rn)to an arbitrary closed subset ofRn.The similar result is obtained for some other spaces of multivariate smooth functions. We also show that unlike the one-dimensional case treated by Whitney, for some trace spaces of multivariate smooth functions a linear bounded extension operator does not exist. The proofs are based on a relation between the problem under consideration and a similar problem for Lipschitz spaces defined on hyperbolic Riemannian manifolds.

Whitney’s extension problem for multivariate ^{1,}-functions

We prove that the trace of the space C1,ω(Rn) to an arbitrary closed subset X ⊂ Rn is characterized by the following “finiteness” property. A function f : X → R belongs to the trace space if and only if the restriction f |Y to an arbitrary subset Y ⊂ X consisting of at most 3·2n−1 can be extended to a function fY ∈ C1,ω(Rn) such that sup{‖fY ‖C1,ω : Y ⊂ X, cardY ≤ 3 · 2n−1} <∞. The constant 3 · 2n−1 is sharp. The proof is based on a Lipschitz selection result which is interesting in its own right. 1. Main results The results of the paper are concerned with the following problem having its origin in two classical papers of Hassler Whitney [W1], [W2] which appeared in 1934. Let C(R) be the space of k-times continuously differentiable functions f satisfying

On extensions of Sobolev functions defined on regular subsets of metric measure spaces

We characterize the restrictions of first-order Sobolev functions to regular subsets of a homogeneous metric space and prove the existence of the corresponding linear extension operator.

Lipschitz selections of set-valued mappings and Helly’s theorem

AbstractWe prove a Helly-type theorem for the family of all m-dimensional convex compact subsets of a Banach space X. The result is formulated in terms of Lipschitz selections of set-valued mappings from a metric space (M, ρ) into this family.Let M be finite and let F be such a mapping satisfying the following condition: for every subset M′ ⊂ M consisting of at most 2m+1 points, the restriction F¦M′ of F to M′ has a selection fM′ (i. e., fM′(x) ∈ F(x) for all x ∈ M′) satisfying the Lipschitz condition ‖ƒM′(x) − ƒM′(y)‖X ≤ ρ(x, y), x, y ∈ M′. Then F has a Lipschitz selection ƒ: M → X such that ‖ƒ(x) − ƒ(y)‖X ≤ γρ(x,y), x, y ∈ M where γ is a constant depending only on m and the cardinality of M. We prove that in general, the upper bound of the number of points in M′, 2m+1, is sharp. If dim X = 2, then the result is true for arbitrary (not necessarily finite) metric space. We apply this result to Whitney’s extension problem for spaces of smooth functions. In particular, we obtain a constructive necessary and sufficient condition for a function defined on a closed subset ofR2to be the restriction of a function from the Sobolev space W∞2(R2).A similar result is proved for the space of functions onR2satisfying the Zygmund condition.

Rearrangement-Function Inequalities and Interpolation Theory

Using the tools of Real Interpolation Theory, we develop a general method for proving rearrangement-function inequalities for important classes of operators.

Barycentric selectors and a Steiner-type point of a convex body in a Banach space

Let F be a mapping from a metric space (M,ρ) into the family of all m-dimensional affine subsets of a Banach space X. We present a Helly-type criterion for the existence of a Lipschitz selection f of the set-valued mapping F, i.e., a Lipschitz continuous mapping f:M→X satisfying f(x)∈F(x),x∈M. The proof of the main result is based on an inductive geometrical construction which reduces the problem to the existence of a Lipschitz (with respect to the Hausdorff distance) selector SX(m) defined on the family Km(X) of all convex compacts in X of dimension at most m. If X is a Hilbert space, then the classical Steiner point of a convex body provides such a selector, but in the non-Hilbert case there is no known way of constructing such a point. We prove the existence of a Lipschitz continuous selector SX(m):Km(X)→X for an arbitrary Banach space X. The proof is based on a new result about Lipschitz properties of the center of mass of a convex set.

A geometrical approach to theK-divisibility problem

We give a geometrical interpretation of the Brudnyi-KrugljakK-divisibility theorem—one of the fundamental results of modern interpolation theory of Banach spaces. We show that this result is closely connected with a curious intersection theorem which can be formulated in the spirit of Helly’s classical theorem. LetB0,B1 be two closed convex balanced subset of a Banach spaceX. We prove that under a wide range of various conditions the family of setsB = {B =sB0 +tB1 +c;s, t ∈R,c ∈X} possesses the following intersection property:LetB′ be a subfamily ofB such that every two sets fromB′ have a common point. Then ∩B∈B′ γ oB ≠ 0, where γ>0 is an absolute constant (γ ≤ 7 + 4 √2) and the symbol γ oB denotes a dilation ofB with respect to its center by a factor of γ.As a consequence we obtain a generalization of theK-divisibility theorem for sums of two elements.

Sharp Finiteness Principles For Lipschitz Selections

Let

On Sobolev extension domains in Rn

{(\mathcal{M}, \rho) }