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 CalderónZygmund singular integrals and their commutators on weighted Morrey spaces and Sobolev spaces.To be precise,letρ>2 and K be a standard Calderón-Zygmund kernel.Denote by V_(ρ)(T_(K))and V_(ρ)(T_(K,b)^(m))(m≥1)theρ-variation operators of Calderón-Zygmund singular integrals and their m-th iterated commutators,respectively.By assuming that V_(ρ)(T_(K))satisfies an a priori estimate,i.e.,the map V_(ρ)(T_(K)):L^(p0)(R^(n))→L^(p0)(R^(n))is bounded for some p0∈(1,∞),the bounds for V_(ρ)(T_(K))and V_(ρ)(T_(K,b)^(m))on weighted Morrey spaces and Sobolev spaces are established.Meanwhile,the compactness properties of V_(ρ)(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.