Estimates for constants in additivity inequalities for function spaces
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 \[ \mu_2(R_1,R_2)=\nu(R_1,R_2)/\text{diam }\Delta\geq\delta, \] one has the inequality \[ \langle f\rangle_{Z(\Omega\cap(R_1\cup R_2))}\leq c_4(\langle f\rangle_{Z(\Omega\cap R_1)}+\langle f\rangle_{Z(\Omega\cap R_2)}), \] then for any \(\gamma>0\), there is a \(c_5(\gamma)>0\) such that for all integers \(m\) and bounded parallelepipeds \(U_1,\dots,U_m\), with faces parallel to the coordinate planes and for which every point of \(\Omega\) has a ball of radius \(\gamma\text{diam }\Omega\) which is contained in some \(U_j\), then one has \[ \langle f\rangle_{Z(\Omega)}\leq c_5\sum^m_{k=1}\langle f\rangle_{Z(\Omega\cap U_k)}. \] They apply this result to study the Sobolev-Slobodetskij spaces \(W^\ell_p(\Omega)\) and \(w^\ell_p(\Omega)\) and deduce, for example, that if \(U_i\) are open parallelepipeds covering \(\overline\Omega\), there is a constant \(c_6(n,\ell,p,U_1,\dots,U_m)>0\) such that \[ |f|_{W^\ell_p(\Omega)}\leq c_6\sum^m_{i=1}|f|_{W^\ell_p(\Omega\cap U_i;\Omega)}. \] They prove the same result for \(w^\ell_p(\Omega)\) if \(\Omega\) is bounded or if the parallelepipeds satisfy the stronger condition in Theorem 2. They also prove lower bounds provided the covering is of multiplicity \(\beta\) (a point belongs to no more than \(\beta\) of the sets \(U_i\)) with bounds depending on \(\beta\) in addition to the other parameters.
scientific article; zbMATH DE number 794613