General theory of global smoothness and approximation by smooth singular operators

In this article we continue with the study of smooth general singular integral operators over the real line regarding their simultaneous global smoothness preservation property with respect to the LpLp norm, 1≤p≤∞1≤p≤∞, by involving higher order moduli of smoothness. Also we study their simultaneous approximation to the unit operator with rates involving the modulus of smoothness. The produced Jackson type inequalities are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function. We finish with applications to trigonometric singular integral operators.