In this chapter, we study the type of Picard singular integral operators on ℝn constructed by means of the nonisotropic β-distance and the q-exponential functions. The central role here is played by the concept of nonisotropic β-distance, which allows us to improve and generalize the results given for classical Picard and q-Picard singular integral operators. In order to obtain the rate of ... [Show full abstract] convergence we introduce a modulus of continuity depending
on the nonisotropic β-distance with respect to the uniform norm. Then we give the definition of β -Lebesgue points depending on nonisotropic β-distance and a pointwise approximation result shown at these points. Futhermore, we present the global smoothness preservation
property of these type of Picard singular integral operators and prove a sharp inequality. This chapter relies on .