On Equivalence of Anchored and ANOVA Spaces; Lower Bounds

Abstract

We provide lower bounds for the norms of embeddings between $\boldsymbolγ$-weighted Anchored and ANOVA spaces of $s$-variate functions with mixed partial derivatives of order one bounded in $L_p$ norm ($p\in[1,\infty]$). In particular we show that the norms behave polynomially in $s$ for Finite Order Weights and Finite Diameter Weights if $p>1$, and increase faster than any polynomial in $s$ for Product Order-Dependent Weights and any $p$.