Lp-general approximations by multivariate singular integral operators

In this article, we study the Lp, 1 ≤ p < ∞ approximation properties of general multivariate singular integral operators over RN, N ≥ 1. We establish their convergence to the unit operator with rates. The established inequalities involve the multivariate higher order modulus of smoothness. We list the multivariate Picard, Gauss–Weierstrass, Poisson Cauchy and trigonometric singular integral operators where this theory can be applied directly.