On equivalence of weighted anchored and ANOVA spaces of functions with mixed smoothness of order one in \(L_1\) or \(L_\infty\) (Q895981)

If \(n\) is a positive integer, let \(\mathbb{R}^n\) denote the Euclidean \(n\)-dimensional space consisting of \(n\)-tuples of real numbers, and let \([1;n]\) denote the set \(\{1,2,\dots,n\}\). If \(u\subset [1;n]\), with complement \(u^\sim=[1;n]\setminus u\), then \(x_u\) denotes \(\{x_a,x_b,\dots, x_c\}\), \(u= \{a,b,\dots,c\}\subseteq [1;n]\), and for a function \(f\) with continuous derivatives, \(f^{(u)}\) denotes \(\prod_{j\in u}(\partial/\partial x_j)f\). For \(1\leq p<\infty\), the space \(L^p(A)\), \(A\subseteq\mathbb{R}^n\), is defined by \(L^p= \{[f]:\|[f]\|_p= \| f\|_p<\infty\}\), where \([f]\) denotes the equivalence class of Lebesgue measurable functions which differ from \(f\) on sets of measure \(0\) and \[ \| f\|_p= \Biggl(\int_A |f(x)|^p dx\Biggr)^{1/p}. \] The Sobolev space \(W^{(1,\dots, 1)}_p(A)= W^{[1;d]}_p(A)\) is defined by \(W^{[1,\dots, 1]}_p(A)= \{f:\| f\|_{S,p}<\infty\}\), where \(\| f\|_{S,p}= \max_{u\subset [1;n]}\| f^{(u)}\|_p<\infty\). The anchored, \(f_\mu\), and ANOVA, \(f_A\), components of a function \(f\) are shown in statements of the paper to have representations of the form \[ \begin{aligned} f_{\mu,u}(x) &= \int_{\mathbb{R}^u} f^{(u)}(0_{u\sim}; t_u)\,K_{\mu, u}(x_u; t_u)\,dt_u;\quad K_{\mu;u}(x_u; t_n)= \prod_{j\in u}\chi_{(0,x_j)}(t_j);\\ f_{A,u}(x) &= \int_{\mathbb{R}^u} \int_{\mathbb{R}^{u\sim}} f^{(u)} (t_{u\sim}, t_u)\,dt_u K_{A; u}(x_u; t_u)\,dt_u;\end{aligned} \] \[ \begin{gathered} K_{A;u} (x_u; t_u)= \prod_{j\in u}(\chi_{(0,x_j)} (t_j)- (1- t_j)),\quad\text{so that}\\ f= \sum_{u\subseteq [1;u]} f_{\mu,u}= \sum_{u\subseteq [1,n]} f_{A;u}.\end{gathered} \] If \(\gamma=\{\gamma_u,u\subset [1;n]\}\), then the \(g\)-weighted anchored and ANOVA spaces have norms defined, respectively, as \[ \begin{aligned} \| f\|_{\mu,p,\gamma}= \left (\sum(1/\gamma^\sim)\| f^{(u)} (.,0_{u^c})\|^p_p \right )^{1/p},\\ \| f\|_{A,p,g}= \left (\sum(1/\gamma^\sim)\| \int_{\mathbb{R}^{u\sim}} f^{(u)}(.; t_u)\, dt_{u^\sim}\|^p_p\right )^{1/p}. \end{aligned} \] The main conclusions indicate conditions under which the weighted norms are equivalent.