Multivariate polynomial inequalities with respect to doubling weights and A∞ weights

In one-dimensional case, various important, weighted polynomial inequalities, such as Bernstein, Marcinkiewicz–Zygmund, Nikolskii, Schur, Remez, etc., have been proved under the doubling condition or the slightly stronger A∞ condition on the weights by Mastroianni and Totik in a recent paper [G. Mastroianni, V. Totik, Weighted polynomial inequalities with doubling and A∞ weights, Constr. Approx. 16 (1) (2000) 37–71]. The main purpose of this paper is to prove multivariate analogues of these results. We establish analogous weighted polynomial inequalities on some multivariate domains, such as the unit sphere Sd−1, the unit ball Bd, and the general compact symmetric spaces of rank one. Moreover, positive cubature formulae based on function values at scattered sites are established with respect to the doubling weights on these multivariate domains. Some of these multi-dimensional results are new even in the unweighted case. Our proofs are based on the investigation of a new maximal function for spherical polynomials.