On traces of functions in abstract spaces of L. N. Slobodetskij (Q1920811)

Let \(A\) be a generator of an analytic semigroup \(e^{-tA}\) with exponentially decaying norm on a Banach space \(E\). The author introduces the abstract Slobodetskij space as a completion, with respect to the norm \[ |v |^p_{W_p^{m+\alpha}(\mathbb R^n_+,E)}= \sum_{|k|+\beta=m}\left(|{\mathcal A}(\beta,k)v|^p_{L_p(\mathbb R^n_+,E)}\int_{\mathbb R^n_+}\int_{\mathbb R^n_+}d^p_{\alpha,(\beta,k)} v(x;y)dxdy\right), \] of the set of infinitely differentiable functions with compact support in \(\mathbb R^n_+\) whose values belong to \(D(A^\infty)\). Here \(\mathcal A (\beta,k)= A^{\beta}D^k,\;D^k={\partial^{k_1}\over\partial x_1^{k_1}}\dots {\partial^{k_n}\over\partial x_n^{k_n}}\), \( k=(k_1,\dots,k_n),\) \(d^p_{\alpha,(\beta,k)}v(x;y)=|x-y|^{-n-\alpha p}|\mathcal A(\beta,k) v(x)-\mathcal A(\beta,k)v(y)|^p\;(0<\alpha<1,p\geq 1).\) The author studies traces of functions from \(W^{m+\alpha}_p(\mathbb R^n_+,E)\) on the hyperplane \(x_n=0\). One of the results of the paper asserts that if \(f\in W_p^{m+\alpha}(\mathbb R^n_+,E)\) with \(\beta +|k|=m-1\), \(m\geq 1\) and \(\alpha p<1\) then \(\mathcal A(\beta,k)f(x_1,\dots,x_{n-1},0)\in W_p^{1+\alpha-1/p}(\mathbb R^{n-1},E)\).