Equivalence of weighted anchored and ANOVA spaces of functions with mixed smoothness of order one in Lp.

We consider $\gamma$-weighted anchored and ANOVA spaces of functions with mixed first order partial derivatives bounded in a weighted $L_p$ norm with $1\leq p\leq infty$. The domain of the functions $D^d$ is , where $D\sbe\mathbbb R$ is a bounded or unbounded interval. We provide conditions on the weights $\gamma$ that guarantee that anchored and ANOVA spaces are equal (as sets of functions) and have equivalent norms with equivalence constants uniformly or polynomially bounded in $d$. Moreover, we discuss applications of these results to integration and approximation of functions on $D^d$.