On extension of Calderon-Zygmund type singular integrals and their commutators

Motivated by the recent works [1, 23], we study the following extension of Calderon-Zygmund type singular integrals T(beta)f(x) = p.v. integral(n)(R) Omega(y)/|y|(n-beta) f(x - y)dy, for 0 < beta < n, and their commutators. We establish estimates of these singular integrals on Lipschitz spaces, Hardy spaces and Muckenhoupt A(p)-weighted L-p-spaces. We also establish Lebesgue and Hardy space estimates of their commutators. Our estimates are uniform in small beta, and therefore one can pass onto the limits as beta -> 0 to deduce analogous estimates for the classical Calderon-Zygmund type singular integrals and their commutators.