Error estimates in $S_p$ for cubature formulas exact for Haar polynomials in the two-dimensional case

On the spaces $S_p$, an upper estimate is found for the norm of the error functional $\delta_N(f)$ of cubature formulas possessing the Haar $d$-property in the two-dimensional case. An asymptotic relation is proved for $\|\delta_N(f)\|_{S_p^*}$ with the number of nodes $N\sim 2^d$, where $d\to\infty$. For $N\sim 2^d$ with $d\to\infty$, it is shown that the norm of $\delta_N$ for the formulas under study has the best convergence rate, which is equal to $N^{-1/p}$.