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We consider $\gamma$-weighted anchored and ANOVA spaces of functions withmixed 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 theweights $\gamma$ that guarantee that anchored and ANOVA spaces are equal (assets of functions) and have equivalent norms with equivalence constantsuniformly or polynomially bounded in $d$. Moreover, we discuss applications ofthese results to integration and approximation of functions on $D^d$.

Equivalence of Weighted Anchored and ANOVA Spaces of Functions with Mixed Smoothness of Order one in $L_p$