On equivalence of weighted anchored and ANOVA spaces of functions with mixed smoothness of order one in L1L1 or L∞L∞

We consider weighted anchored and ANOVA spaces of functions with mixed first order partial derivatives bounded in L1L1 or L∞L∞ norms. We provide conditions on the weights under which the corresponding spaces have equivalent norms with constants independent of, or only polynomially dependent on the number of variables.