Dimension free estimates for the oscillation of Riesz transforms (Q1885652)

Let \(\{T_r\}_r\) be a family of operators in \(L^p\) such that the limit \(Tf=\lim_{r\to 0} T_r\) exists in some sense. Then one can define the oscillation operator \(O\) and the \(\rho\)-variation operator \(V_\rho\), in order to measure the speed of convergence of the family \(\{T_r\}_r\). In this paper, the authors study the oscillation and variation of the Riesz transforms \(R_j\) in the weighted Lebesgue space \(L^p(| x| ^{\alpha})\), proving that if \(1<p<\infty\) and \(-1<\alpha<p-1\) and \(\rho>2\), \[ \| O(R_jf)\| _{L^p(| x| ^{\alpha})}\leq C_{\alpha} \| f\| _{L^p(| x| ^{\alpha})} \] and \[ \| V_{\rho}(R_jf)\| _{L^p(| x| ^{\alpha})}\leq C_{\alpha,\rho} \| f\| _{L^p(| x| ^{\alpha})}, \] where both \(C_{\alpha}\) and \(C_{\alpha,\rho}\) are independent of \(n\). In fact, they obtain the boundedness of those operators on \(L^p(u)\) where \(u\) is a weight in the Muckenhoupt class \(A_p\). They also prove some weighted transference result for positive operators induced by flows on \(L^p\). This transference technique allows them to extend the above result to the \(\ell^2(\{1,2\cdots, n\})\)-valued operator \(Rf(x)=(R_1 f(x), \cdots R_nf(x))\).