Generalized Picard singular integrals

In this article, we introduce and study a new type of Picard singular integral operators on RnRn constructed by means of the concept of the nonisotropic ββ-distance and the qq-exponential functions. The central role here is played by the concept of nonisotropic ββ-distance, which allows one to improve and generalize the results given for classical Picard and qq-Picard singular integral operators. In order to obtain the rate of convergence we introduce a new type of modulus of continuity depending on the nonisotropic ββ-distance with respect to the uniform norm. Then we give the definition of ββ-Lebesque points depending on nonisotropic ββ-distance and a pointwise approximation result shown at these points. Furthermore, we study the global smoothness preservation property of these new type Picard singular integral operators and prove a sharp inequality.