- 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.
- 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)
- 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.
- 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.
- 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.
- 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
- 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.
- 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.
- Sharp L^p estimates for second order discrete Riesz transforms,
Adv. Math. 262 (2014), 932--952
joint with K. Domelevo
We give the optimal Lp bound for second order discrete 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
- 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.
- 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.
- Differential subordination under a change of law,
preprint. joint with K. Domelevo
It is proved that continuous indexed martingales under weak assumptions and in particular no continuity on the path, are bouded in L2 after changing the law in accordance with the usual A2 characteristic. The proof includes the construction of a closed expression of a single Bellman function (in the weak form) of four variables for the weighted problem. The proof also contains an interesting ideological component in its use of the so-called ellipse lemma to pass from a weak type function to the use of strong type subordination condition of martingales. This is an observation that appears only to be needed when the time index is continuous.
- Convex body domination and weighted estimates with matrix weight,
Adv.Math. (2017) joint with F. Nazarov, S. Treil, A. Volberg
We use convex bodies to give a sparse domination formula for vector valued operators. From this, we deduce the best to date estimate for Calderon-Zygmund operators with matrix weight.
- Weighted little BMO and two-weight inequalities for Journe commutators,
Analysis PDE (2017) joint with I. Holmes, B. Wick
This is the Bloom theory of commutators in the bi-parameter setting. The proof includes a treatment of weighted estimates of Journe operators different from Fefferman's (difficult) proof, simpler, with modern tools.
- On the failure of lower square function estimates in the non-homogenous weighted setting,
preprint joint with K. Domelevo, P. Ivanisvili, S. Treil, A. Volberg
Some surprising negative results on the lower square function estimates with weight.
- The sharp square function estimate with matrix weight,
preprint joint with T. Hytonen, A. Volberg
Finally the first sharp estimate of a singular operator with matrix weight.
10 Selected Pulications
You can get my published results on mathscinet or arxiv.