Estimates for constants in additivity inequalities for function spaces
type: Item
instance of: scholarly article
title: Estimates for constants in additivity inequalities for function spaces
zbMATH Open document ID: 0863.46024
DOI: 10.1007/BF02104943
author: A. Senusi
author: Victor I. Burenkov
published in: Siberian Mathematical Journal
publication date: 1995-09-18T00:00:00Z
review text: The authors study geometric conditions on domains \(U_i\) that give estimates for the constants in embedding theorems, specifically investigating the dependence on the constant associated with a union of domains on the constants for the individual domains. For the general problem, let \(\Omega\) be an open subset of \(\mathbb{R}^n\) with \(G\) an open subset of \(\Omega\) and suppose that \(Z(G)\) is a space of functions defined on \(G\) with an associated functional \(\langle f\rangle_{Z(G)}\). Then \(Z(G)\) is additive with respect to open sets \(G_1,\dots,G_m\) if the fact that a function defined on \(\bigcup^m_{k=1}G_k\) belongs to \(\bigcap^m_{k=1}Z(G_k)\) implies that the function belongs to \(Z(\bigcup^m_{k=1}G_k)\) and \[ \langle f\rangle_{Z(\bigcup^m_{k=1}G_k)}\leq c_3\sum^m_{k=1} \langle f\rangle_{Z(G_k)}, \] where \(c_3\) is a constant independent of \(f\). A family of spaces has the monotonicity property if for arbitrary open subsets \(G_1\), \(G_2\) of \(\Omega\), with \(G_1\subseteq G_2\), we have \[ \langle f\rangle_{Z(G_1)}\leq\langle f\rangle_{Z(G_2)}, \] for all \(f\) in \(Z(G_2)\). If \(R_1\), \(R_2\) are open parallelepipeds with faces parallel to the coordinate planes, which do not contain one another and have a common projection \(\Delta\) onto one of the coordinate planes, for instance, on the plane \(x_j=0\), let \(\nu(R_1,R_2)=\text{dist}(R_1\backslash R_2,R_2\backslash R_1)\). Let \(\ell_j(R_1)\) and \(\ell_j(R_2)\) denote the lengths of the edges of \(R_1\) and \(R_2\) that are parallel to the \(Ox_j\) axis. For any set \(X\), let \(X_\delta=\{x \mid\text{dist}(x,X)\geq\delta\}\). The authors' main general result (Theorem 2) is that if \(\Omega\) is as above and the spaces \(Z(G)\) have the monotonicity property, and for each \(\delta\in(0,1)\), there is a \(c_4(\delta)>0\) such that for all mentioned parallelepipeds with \[ \mu_1(R_1,R_2)=\nu(R_1,R_2)/\min((\ell_j(R_1),\ell_j(R_2))\geq\delta, \] and \[ \mu_2(R_1,R_2)=\nu(R_1,R_2)/\text{di…
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 794613
zbMATH Keywords: geometric conditions on domains
zbMATH Keywords: embedding theorems
zbMATH Keywords: monotonicity property
zbMATH Keywords: Sobolev-Slobodetskij spaces
reviewed by: Raymond Johnson
MaRDI profile type: Publication
cites work: Q3825006
cites work: Q5719719
cites work: Q4711784
cites work: Reduction of nonholonomic variation problems to isoperimetric ones and connections in principal bundles
cites work: On a constrained variational problem and the space of horizontal paths
full work available at URL: https://doi.org/10.1007/bf02104943
OpenAlex ID: W2046838141
description: scientific article; zbMATH DE number 794613