Some fragments of second-order logic over the reals for which satisfiability and equivalence are (un)decidable

We consider the $\Sigma_0^1$-fragment of second-order logic over the vocabulary $\langle+, \times, 0, 1, <, S_1, ..., S_k \rangle$, interpreted over the reals, where the predicate symbols $S_i$ are interpreted as semialgebraic sets. We show that, in this context, satisability of formulas is decidable for the first-order $\exists$*-quantifier fragment and undecidable for the $\exists$*$\forall$- and $\forall$*-fragments. We also show that for these three fragments the same (un)decidability results hold for containment and equivalence of formulas.