10 Selected Pulications

 

You can get my published results on mathscinet or arxiv.

  1. Dyadic Shifts and a Logarithmic Estimate for Hankel Operators with Matrix Symbol,
    Comptes Rendus Acad. Sci. Paris, t.330, no.1, pp.455-460, 2000.
    Contains a surprising fact: the Hilbert transform is a coefficent shift/multiplier operator on the Haar wavelet system. As a corollary, a question on commutators with matrix symbols is solved.

  2. An Estimate for Weighted Hilbert Transform via Square Functions,
    Trans. Amer. Math. Soc 354, 2002, pp.1699-1703.
    joint with S. Pott
    Hunt-Muckenhoupt-Wheeden in one line, with best to date bounds. A consequence of 1)

  3. Heating of the Beurling Operator: Weakly Quasiregular Maps on the Plane are Quasiregular,
    Duke Math. J. Vol. 112, No 2, 2002, pp.281-305.
    joint with A. Volberg
    An old borderline regularity conjecture that was brought to us by Kari Astala and that had been reduced to a sharp weighted inequality. It also contains the first A_2 theorem, for the Beurling operator. It contains a new identity formula using heat equation.

  4. The Sharp Bound for the Hilbert Transform on Weighted Lebesgue Spaces in Terms of the Classical A_p Characteristic,
    Amer. J. Math. 129 (2007), no. 5, 1355--1375.
    The solution of the A_2 theorem for the Hilbert transform. A not very direct consequence of 1). The Haar shift operators do not want to be estimated by Bellman function (quote of the anonymous referee) Indeed, a bilinear estimate of time shifted carée du champ expressions is key to the estimate.

  5. A Rotation Method which Gives Linear L^p-Estimates for Powers of the Ahlfors-Beurling Operator,
    J. Math. Pures Appl. (9) 86, 2006, no. 6, 492--509.
    joint with O. Dragicevic and A. Volberg
    The title says it all.

  6. A p-adapted Square function and the L^p Dirichlet problem,
    J. Funct. Anal. 249, 2007, no. 2, pp.372--392
    joint with M. Dindos and J. Pipher
    Solvability for small p under additional assumptions, the square function is replaced by its p-adapted variant and lends itself perfectly to integration by parts

  7. Higher order Riesz commutators,
    Amer. J. Math. 131 (2009), no. 3, 731--769.
    joint with M. Lacey, J. Pipher and B. Wick
    A classification result of product BMO via iterated commutators in several variables. Technically demanding.

  8. Sharp A_2 inequality for Haar Shift Operators,
    Math. Ann. 348 (2010), no. 1, 127--141.
    joint with M. Lacey and M. Reguera
    Finally a non-Bellman proof of 4) for those who are still not loving it.

  9. Sharp L^p estimates for second order discrete Riesz transforms,
    Adv. Math. 262 (2014), 932--952
    joint with K. Domelevo
    The sharp L^p estimate for the discrete Hilbert transform (on the integers) is a very deep long standing open question. We resolve this question for second order Riesz transforms. It is the first sharp estimate of any discrete Calderon-Zygmund operator. This first proof is deterministic, but was followed by a probabilistic argument by the authors and additional applications

  10. Higher order Journé commutators and characterizations of multi-parameter BMO,
    Adv. Math. 291 (2016) 24--58. joint with Y. Ou and E. Strouse
    A natural endpoint to a series of deep papers on characterization of multi-parameter BMO spaces through boundedness of commutators of Hilbert or Riesz transforms and symbol functions.

  11. Sharp weighted norm estimates beyond Calderon-Zygmund theory,
    Anal. PDE 9 (2016) 1079--1113. joint with F. Bernicot and D. Frey
    We bring sparse domination to a wide range of operators, not necessarily bounded in all L^p and derive optimal weighted estimate for admissible ranges of p in terms of the Auscher-Martell characteristic. Ideas in this proof streamline the argument previously used for Calderon-Zygmund operators through the use of an adapted maximal operator.