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Pavel Shvartsman

Publications and source records attributed to Pavel Shvartsman.

At least 19 recordsLinked to original sources
arXiv2026

Sharp Finiteness Principles for the boundary values of $C^2$-functions: long version

Let $Ω$ be a domain in ${\bf R}^n$ with boundary $\partialΩ$. We prove the following Finiteness Principle for the boundary values of $C^2(Ω)$-functions: A function $f:\partialΩ\to {\bf R}$ is the trace to the boundary of a function $F\in C^2(Ω)$ provided there exists a constant $λ>0$ such that for every set $E\subset\partialΩ$ consisting of at most $N=3\cdot 2^{n-1}$ points, there exists a function $F_E\in C^2(Ω)$ with $\|F_E\|_{C^2(Ω)}\leλ$ whose trace to $\partialΩ$ coincides with $f$ on $E$. Furthermore, we refine this principle by showing that this criterion can be restricted to $N$-point subsets $E\subset\partialΩ$ that possess specific geometric ``visibility'' properties relative to $Ω$.

math.FA↗
arXiv2025

Efficient Algorithms for Lipschitz Selections of Set-Valued Mappings in ${\bf R}^2$: long version

Let $F$ be a set-valued mapping from an $N$-element metric space $({\mathcal M},ρ)$ into the family of all closed half-planes in ${\bf R}^2$. In this paper, we provide an efficient algorithm for a Lipschitz selection of $F$, i.e., a Lipschitz mapping $f:{\mathcal M}\to{\bf R}^2$ such that $f(x)\in F(x)$ for all $x\in{\mathcal M}$. Given a constant $λ>0$, this algorithm produces the following two outcomes: (1) The algorithm guarantees that there is no Lipschitz selection of $F$ with Lipschitz constant at most $λ$; (2) The algorithm returns a Lipschitz selection of $F$ with Lipschitz constant at most $3λ$. The total work and storage required by this selection algorithm are at most $CN^2$ where $C$ is an absolute constant.

math.FA↗
arXiv2023

Existence Criteria for Lipschitz Selections of Set-Valued Mappings in ${\bf R}^2$

Let $F$ be a set-valued mapping which to each point $x$ of a metric space $({\mathcal M},ρ)$ assigns a convex closed set $F(x)\subset{\bf R}^2$. We present several constructive criteria for the existence of a Lipschitz selection of $F$, i.e., a Lipschitz mapping $f:{\mathcal M}\to{\bf R}^2$ such that $f(x)\in F(x)$ for every $x\in{\mathcal M}$. The geometric methods we develop to prove these criteria provide efficient algorithms for constructing nearly optimal Lipschitz selections and computing the order of magnitude of their Lipschitz seminorms.

math.FA↗
arXiv2022

On the Core of a Low Dimensional Set-Valued Mapping

Let ${\mathfrak M}=({\mathcal M},ρ)$ be a metric space and let $X$ be a Banach space. Let $F$ be a set-valued mapping from ${\mathcal M}$ into the family ${\mathcal K}_m(X)$ of all compact convex subsets of $X$ of dimension at most $m$. The main result in our recent joint paper with Charles Fefferman (which is referred to as a "Finiteness Principle for Lipschitz selections") provides efficient conditions for the existence of a Lipschitz selection of $F$, i.e., a Lipschitz mapping $f:{\mathcal M}\to X$ such that $f(x)\in F(x)$ for every $x\in{\mathcal M}$. We give new alternative proofs of this result in two special cases. When $m=2$ we prove it for $X={\bf R}^{2}$, and when $m=1$ we prove it for all choices of $X$. Both of these proofs make use of a simple reiteration formula for the "core" of a set-valued mapping $F$, i.e., for a mapping $G:{\mathcal M}\to{\mathcal K}_m(X)$ which is Lipschitz with respect to the Hausdorff distance, and such that $G(x)\subset F(x)$ for all $x\in{\mathcal M}$.

math.FA↗
arXiv2021

The Core of a 2-Dimensional Set-Valued Mapping. Existence Criteria and Efficient Algorithms for Lipschitz Selections of Low Dimensional Set-Valued Mappings

Let ${\mathfrak M}=({\mathcal M},ρ)$ be a metric space and let $X$ be a Banach space. Let $F$ be a set-valued mapping from ${\mathcal M}$ into the family ${\mathcal K}_m(X)$ of all compact convex subsets of $X$ of dimension at most $m$. The main result in our recent joint paper with Charles Fefferman (which is referred to as a "Finiteness Principle for Lipschitz selections") provides efficient conditions for the existence of a Lipschitz selection of $F$, i.e., a Lipschitz mapping $f:{\mathcal M}\to X$ such that $f(x)\in F(x)$ for every $x\in{\mathcal M}$. We give new alternative proofs of this result in two special cases. When $m=2$ we prove it for $X={\bf R}^{2}$, and when $m=1$ we prove it for all choices of $X$. Both of these proofs make use of a simple reiteration formula for the "core" of a set-valued mapping $F$, i.e., for a mapping $G:{\mathcal M}\to{\mathcal K}_m(X)$ which is Lipschitz with respect to the Hausdorff distance, and such that $G(x)\subset F(x)$ for all $x\in{\mathcal M}$. We also present several constructive criteria for the existence of Lipschitz selections of set-valued mappings from ${\mathcal M}$ into the family of all closed half-planes in ${\bf R}^{2}$.

math.FA↗
arXiv2019

Sobolev functions on closed subsets of the real line

For each $p>1$ and each positive integer $m$ we use divided differences to give intrinsic characterizations of the restriction of the Sobolev space $W^m_p(R)$ to an arbitrary closed subset of the real line.

math.FA↗
arXiv2018

Sobolev functions on closed subsets of the real line: long version

For each $p>1$ and each positive integer $m$ we give intrinsic characterizations of the restriction of the Sobolev space $W^m_p(R)$ and homogeneous Sobolev space $L^m_p(R)$ to an arbitrary closed subset $E$ of the real line. In particular, we show that the classical one dimensional Whitney extension operator is "universal" for the scale of $L^m_p(R)$ spaces in the following sense: for every $p\in(1,\infty]$ it provides almost optimal $L^m_p$-extensions of functions defined on $E$. The operator norm of this extension operator is bounded by a constant depending only on $m$. This enables us to prove several constructive $W^m_p$- and $L^m_p$-extension criteria expressed in terms of $m^{th}$ order divided differences of functions.

math.FA↗
arXiv2018

Extension criteria for homogeneous Sobolev space of functions of one variable

For each $p>1$ and each positive integer $m$ we give intrinsic characterizations of the restriction of the homogeneous Sobolev space $L^m_p(R)$ to an arbitrary closed subset $E$ of the real line. We show that the classical one dimensional Whitney extension operator is "universal" for the scale of $L^m_p(R)$ spaces in the following sense: for every $p\in(1,\infty]$ it provides almost optimal $L^m_p$-extensions of functions defined on $E$. The operator norm of this extension operator is bounded by a constant depending only on $m$. This enables us to prove several constructive $L^m_p$-extension criteria expressed in terms of $m^{th}$ order divided differences of functions.

math.FA↗
arXiv2018

Sharp finiteness principles for Lipschitz selections

Let $(M,ρ)$ be a metric space and let $Y$ be a Banach space. Given a positive integer $m$, let $F$ be a set-valued mapping from $M$ into the family of all compact convex subsets of $Y$ of dimension at most $m$. In this paper we prove a finiteness principle for the existence of a Lipschitz selection of $F$ with the sharp value of the finiteness constant.

math.FA↗
arXiv2017

Sharp finiteness principles for Lipschitz selections: long version

Let $({\mathcal M},ρ)$ be a metric space and let $Y$ be a Banach space. Given a positive integer $m$, let $F$ be a set-valued mapping from ${\mathcal M}$ into the family of all compact convex subsets of $Y$ of dimension at most $m$. In this paper we prove a finiteness principle for the existence of a Lipschitz selection of $F$ with the sharp value of the finiteness number.

math.FA↗
arXiv2016

Whitney-type extension theorems for jets generated by Sobolev functions

Let $L^m_p(R^n)$, $p\in [1,\infty]$, be the homogeneous Sobolev space, and let $E\subset R^n$ be a closed set. For each $p>n$ and each non-negative integer $m$ we give an intrinsic characterization of the restrictions to $E$ of $m$-jets generated by functions $F\in L^{m+1}_p(R^n)$. Our trace criterion is expressed in terms of variations of corresponding Taylor remainders of $m$-jets evaluated on a certain family of "well separated" two point subsets of $E$. For $p=\infty$ this result coincides with the classical Whitney-Glaeser extension theorem for $m$-jets. Our approach is based on a representation of the Sobolev space $L^{m+1}_p(R^n)$, $p>n$, as a union of $C^{m,(d)}(R^n)$-spaces where $d$ belongs to a family of metrics on $R^n$ with certain "nice" properties. Here $C^{m,(d)}(R^n)$ is the space of $C^m$-functions on $R^n$ whose partial derivatives of order $m$ are Lipschitz functions with respect to $d$. This enables us to show that, for every non-negative integer $m$ and every $p\in (n,\infty)$, the very same classical linear Whitney extension operator provides an almost optimal extension of $m$-jets generated by $L^{m+1}_p$-functions.

math.FA↗
arXiv2015

On planar Sobolev $L^m_p$-extension domains

For each $m\ge 1$ and $p>2$ we characterize bounded simply connected Sobolev $L^m_p$-extension domains $Ω\subset R^2$. Our criterion is expressed in terms of certain intrinsic subhyperbolic metrics in $Ω$. Its proof is based on a series of results related to the existence of special chains of squares joining given points $x$ and $y$ in $Ω$. An important geometrical ingredient for obtaining these results is a new "Square Separation Theorem". It states that under certain natural assumptions on the relative positions of a point $x$ and a square $S\subsetΩ$ there exists a similar square $Q\subsetΩ$ which touches $S$ and has the property that $x$ and $S$ belong to distinct connected components of $Ω\setminus Q$.

math.FA↗
arXiv2013

Lipschitz spaces generated by the Sobolev-Poincaré inequality and extensions of Sobolev functions

Let $d$ be a metric on $R^n$ and let $C^{m,(d)}(R^n)$ be the space of $C^m$-function on $R^n$ whose partial derivatives of order $m$ belong to the space $Lip(R^n;d)$. We show that the homogeneous Sobolev space $L^{m+1}_p(R^n),p>n,$ can be represented as a union of $C^{m,(d)}(R^n)$-spaces where $d$ belongs to a family of metrics on $R^n$ with certain "nice" properties. This enables us in several important cases to give intrinsic characterizations of the restrictions of Sobolev spaces to arbitrary closed subsets of $R^n$. In particular, we generalize the classical Whitney extension theorem for the space $C^m(R^n)$ to the case of the Sobolev space $L^m_p(R^n)$ whenever $m\ge 1$ and $p>n$.

math.FA↗
arXiv2012

Sobolev $L^2_p$-functions on closed subsets of $R^2$

For each $p>2$ we give intrinsic characterizations of the restriction of the homogeneous Sobolev space $L^1_p(R^2)$ to an arbitrary finite subset $E$ of $R^2$. The trace criterion is expressed in terms of certain weighted oscillations of the second order with respect to a measure generated by the Menger curvature of triangles with vertices in $E$.

math.FA↗
arXiv2012

On The Sum Of A Sobolev Space And A Weighted $L_P$-Space

Let $p>n$ and let $L^1_p(R^n)$ be a homogeneous Sobolev space. For an arbitrary Borel measure $μ$ on $R^n$ we give a constructive characterization of the space $L^1_p(R^n)+L_p(R^n;μ)$. We express the norm in this space in terms of certain oscillations with respect to the measure $μ$. This enables us to describe the $K$-functional for the couple $(L_p(R^n;μ),L^1_p(R^n))$ in terms of these oscillations, and to prove that this couple is quasi-linearizable.

math.FA↗
arXiv2010

A new look at the John-Nirenberg and John-Stromberg theorems for BMO. Lecture Notes

We develop some techniques for studying various versions of the function space BMO. Special cases of one of our results give alternative proofs of the celebrated John- Nirenberg inequality and of related inequalities due to John and to Wik. Our approach enables us to pose a simply formulated "geometric" question, for which an affirmative answer would lead to a version of the John-Nirenberg inequality with dimension free constants. A more detailed summary of the main ideas and results of this paper can be found at http://www.math.technion.ac.il/~mcwikel/bmo/CwikSaghShvaSummary.pdf

math.FA↗
arXiv2009

The Whitney extension problem for Zygmund spaces and Lipschitz selections in hyperbolic jet-spaces

We study a variant of the Whitney extension problem for the space $C^kΛ^m_ω(R^n)$ of functions whose partial derivatives of order $k$ satisfy the generalized Zygmund condition. We identify $C^kΛ^m_ω(R^n)$ with a space of Lipschitz mappings from a metric space $(R^{n+1}_+,ρ_ω)$ supplied with a hyperbolic metric $ρ_ω$ into a metric space $({\cal P}_{k+m-1}\times R^{n+1}_+, d_ω)$ of polynomial fields on $R^{n+1}_+$ equipped with a hyperbolic-type metric $d_ω$. This identification allows us to reformulate the Whitney problem for $C^kΛ^m_ω(R^n)$ as a Lipschitz selection problem for set-valued mappings from $(R^{n+1}_+,ρ_ω)$ into a certain family of subsets of ${\cal P}_{k+m-1}\times R^{n+1}_+$.

math.FA↗