Dimension free estimates for the oscillation of Riesz transforms

In this paper we establish dimension free Lp(Rn, |x|α) norm inequalities (1 < p < ∞) for the oscillation and variation of the Riesz transforms in Rn. In doing so we find Ap−weighted norm inequalities for the oscillation and the variation of the Hilbert transform in R. Some weighted transference results are also proved. INTRODUCTION Throughout (X,F , μ) will denote an arbitrary σ−finite measure space. Let {Tr} be a family of operators bounded from Lp(X,F , μ) into itself for some p in the range 1 < p < ∞ and such the limit Tf = lim r↘0 Trf, for functions f ∈ Lp(X,F , μ), exists in some sense. A classical method of measuring the speed of convergence of the family {Tr} is to consider “square functions” of the type ( ∞ ∑ i=1 |Trif − Tri+1f |2 )1/2 where ri ↘ 0. In the last decade and mainly in the context of ergodic theory, see [JKRW] and the references there, other expressions have been considered to measure the speed of convergence. Let T be an operator such that T = lim r↘0 Tr as above for a family of operators {Tr}r>0. Given {ti}i a fixed decreasing sequence ti ≥ ti+1 ↘ 0 we define the oscillation operator as O(Tf)(x) = ( ∞ ∑ i=1 sup ti+1≤εi+1<εi≤ti |Tεi+1f(x)− Tεif(x)| )1/2 . We shall also consider the operator