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

We consider γ-weighted anchored and ANOVA spaces of functions with mixed first order partial derivatives bounded in a weighted L_p norm with 1≤p≤∞. The domain of the functions is D^d, where D⊆R is a bounded or unbounded interval. We provide conditions on the weights γ 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.