Norm-variation of ergodic averages with respect to two commuting   transformations
  We study double ergodic averages with respect to two general commuting transformations and establish a sharp quantitative result on their convergence in the norm. We approach the problem via real harmonic analysis, using recently developed methods for bounding multilinear singular integrals with certain entangled structure. A byproduct of our proof is a bound for a two-dimensional bilinear square function related to the so-called triangular Hilbert transform. 
Polona Durcik, Vjekoslav Kova\v{c}, Kristina Ana \v{S}kreb, Christoph   Thiele
math.DS math.CA