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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 $$(\| f\|_{p,W(\Omega,\gamma)})^p\leq M_2 N_1(g)$$.
##### MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
##### Keywords:
Lipschitz functions; Sobolev spaces
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##### References:
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