Variation operators for singular integrals and their commutators on weighted Morrey spaces and Sobolev spaces

In this paper, we investigate the boundedness and compactness for variation operators of Calderon-Zygmund singular integrals and their commutators on weighted Morrey spaces and Sobolev spaces. To be precise, let rho > 2 and K be a standard Calderon-Zygmund kernel. Denote by V-rho(T-K) and V-rho(T-K,b(m)) (m >= 1) the rho-variation operators of Calderon-Zygmund singular integrals and their m-th iterated commutators, respectively. By assuming that V-rho(T-K) satisfies an a priori estimate, i.e., the map V-rho(T-K) : L-p0 (R-n) -> L-p0 (R-n) is bounded for some p(0) is an element of (1, infinity), the bounds for V rho(T-K) and V-rho(T-K,b(m)) on weighted Morrey spaces and Sobolev spaces are established. Meanwhile, the compactness properties of V-rho(T-K,b(m)) on weighted Lebesgue and Morrey spaces are also discussed. As applications, the corresponding results for the Hilbert transform, the Hermite Riesz transform, Riesz transforms and rough singular integrals as well as their commutators on the above function spaces are presented.