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Boundary values of functions from Sobolev spaces with Mockenhaupt weight on non-Lipschitz domains. (English. Russian original) Zbl 1326.46031
Dokl. Math. 89, No. 3, 338-342 (2014); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 456, No. 4, 408-412 (2014).
Let \(\mathbb{R}^n\) denote the Euclidean \(n\)-dimensional space for \(n\geq 2\). If \(I\) denotes the interval \((-1,1)\) and \(\mathbb Z\) denotes the set of positive integers, then \(I^n\) and \(\mathbb Z^n\) denote \[ (-1,1)^n= \{x=(x_1, x_2,\dots, x_n):-1< x_j<1,\,j= 1,2,\dots, n\} \] and \(\{m= (m_1,m_2,\dots, m_n): m_j\in \mathbb Z,\,j= 1,2,\dots, n\}\). An open cube \(Q^n\) in \(\mathbb R^n\) and the dyadic open cube \(Q^n_{k,m}\), \(k\in \mathbb Z\), \(m\in \mathbb Z^n\), are represented as \(Q^n= \prod^n_{j=1} (a_j,b_j)\), \((a_j,b_j)\subset \mathbb R\), \(Q^n_{k,m}= \prod^n_{j=1} (2^{-k} m_j, 2^{-k}(m_j+1)\). If \(\gamma\) is nonnegative and Lebesgue measurable on \(\mathbb R^n\), then the weighted \(L^p\)-space \(L^p(E,\gamma)\), \(E\subseteq \mathbb R^n\), with norm \(\|\cdot\|_{p,E(\gamma)}\), is defined by \[ L^p(E,\gamma)= \{[f], \|[f]\|_{p,(E,\gamma)}=\| f\|_{p,(E,\gamma)}<\infty\}, \] where \[ \| f\|_{p,(E,\gamma)}= \Biggl(\int_E\gamma(x)|f(x)|^p\,dx\Biggr)^{1/p}, \] and \([f]\) is the equivalence class of functions which differ from \(f\) on sets of measure \(0\). If \(\nabla=(D_1,D_2,\dots, D_n)\), where \(D_1,D_2,\dots, D_n\), are generalised Sobolev derivatives of functions on a domain \(\Omega\), then \(W^1_p(\Omega,\gamma)\) denotes the Sobolev function space on \(\Omega\) with weighted norm \(\|\cdot\|_{p,W(\Omega,\gamma)}\) defined by \[ \|\cdot\|_{p,W(\Omega,\gamma)}= \| f\|_{p,(\Omega,g)}+ \|\nabla(f)\|_{p,(\Omega,\gamma)}. \] The first main statement of this paper indicates that if \(f\in W^1_p(\Omega,\gamma)\), then there is a trace \(g=\text{tr}|_{\partial\Omega}f\) of \(f\) such that \[ \begin{split} N_1(g)= \sum^\infty_{j=1}\| g_1\|_{p,W(\zeta,G_1)}+\| g_2\|_{p,W(\zeta,G_2)}+ \sum^\infty_{j=1} \varphi(x_j)^{-np} \gamma(P^n_j)\Biggl(\int_J |g_2(y)- g_1(y)|\,dy\Biggr)^p\\ \leq M_1(\| f\|_{p,W(\Omega,\gamma)})^p,\quad J= (1+2\lambda) Q^{n-1}(\varphi),\end{split} \] where the terms \(g_1\), \(g_2\), \(\zeta\), \(G_1\), \(G_2\), are defined in terms of a function \(\varphi\) satisfying the Lipschitz condition and components \(Q^{n-1}(f)\), \(\{\gamma^1_{k,m},\,k,m\geq 1\}\) and \(\{\gamma_{k,m}^2,\, k,m\geq 1\}\).
A second statement indicates that if \(g:\partial\Omega\to \mathbb R\), then \(f\in W^1_p(\Omega,\gamma)\) exists such that \((\| f\|_{p,W(\Omega,\gamma)})^p\leq M_2 N_1(g)\).
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI
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