Positive sparse domination of variational Carleson operators

Due to its nonlocal nature, the r-variation norm Carleson operator Cr does not yield to the sparse domination techniques of Lerner [15, 17], Di Plinio and Lerner [6], Lacey [14]. We overcome this difficulty and prove that the dual form of Cr can be dominated by a positive sparse form involving Lp averages. Our result strengthens the Lp-estimates by Oberlin et al. [18]. As a corollary, we obtain quantitative weighted norm inequalities improving the results in [8] by Do and Lacey. Our proof relies on the localized outer Lp-embeddings of Di Plinio and Ou [7] and Uraltsev [19].