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  • Hanninen, Timo S. (2017)
    In this note, we extend Lerner's local median oscillation decomposition to arbitrary (possibly non-doubling) measures. In the light of the analogy between median and mean oscillation, our extension can be viewed as a median oscillation decomposition adapted to the dyadic (martingale) BMO. As an application of the decomposition, we give an alternative proof for the dyadic (martingale) John-Nirenberg inequality, and for Lacey's domination theorem, which states that each martingale transform is pointwise dominated by a positive dyadic operator of zero complexity. Furthermore, by using Lacey's recent technique, we give an alternative proof for Conde-Alonso and Rey's domination theorem, which states that each positive dyadic operator of arbitrary complexity is pointwise dominated by a positive dyadic operator of zero complexity.
  • Hytönen, Tuomas P. (2017)
    This exposition presents a self-contained proof of the A(2) theorem, the quantitatively sharp norm inequality for singular integral operators in the weighted space L-2 (w). The strategy of the proof is a streamlined version of the author's original one, based on a probabilistic Dyadic Representation Theorem for singular integral operators. While more recent non-probabilistic approaches are also available now, the probabilistic method provides additional structural information, which has independent interest and other applications. The presentation emphasizes connections to the David-Journe T(1) theorem, whose proof is obtained as a byproduct. Only very basic Probability is used; in particular, the conditional probabilities of the original proof are completely avoided. (C) 2016 Elsevier GmbH. All rights reserved.
  • Hänninen, Timo S.; Lorist, Emiel (2019)
    We study the domination of the lattice Hardy-Littlewood maximal operator by sparse operators in the setting of general Banach lattices. We prove that the admissible exponents of the dominating sparse operator are determined by the q-convexity of the Banach lattice.
  • Li, Kangwei (2018)
    In this note, we show that if T is a multilinear singular integral operator associated with a kernel satisfies the so-called multilinear L-r-Hormander condition, then T can be dominated by multilinear sparse operators.
  • Li, Kangwei (2017)
    Let 1 Given a pair of weights and a sparse family , we study the two weight inequality for the following bi-sublinear form
  • Hytönen, Tuomas P.; Li, Kangwei (2018)
    We prove sharp weak and strong type weighted estimates for a class of dyadic operators that includes majorants of both standard singular integrals and square functions. Our main new result is the optimal bound left perpendicular w right perpendicular(Ap)(1/p) left perpendicular w right perpendicular(A infinity)(1/2) (1/p) less than or similar to left perpendicular w right perpendicular(Ap)(1/2) for the weak type norm of square functions on L-p(w) for p > 2; previously, such a bound was only known with a logarithmic correction. By the same approach, we also recover several related results in a streamlined manner.