  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 is $D^d$, where $D \subseteq
\mathbb{R}$ 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 $D^d$.
