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Harmonic Analysis


These are the results obtained under the ERC grant. The papers are available on Arxiv.

Differential subordination under a change of law,

Ann. Prob., 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., 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 Journé commutators

Analysis PDE, 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-homogeneous weighted setting,

Math. Ann., 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,

Discrete Anal., joint with T. Hytonen, A. Volberg

Finally the first sharp estimate of a singular operator with matrix weight. The proof contains a sparse domination of the integrated version of the square function with matrix weight.

Continuous-time sparse domination,

preprint, joint with K. Domelevo

We develop the self similarity argument known as sparse domination in an abstract martingale setting, using a continuous time parameter. With this method, we prove a sharp weighted L^p estimate for the maximal operator Y^* of Y with respect to X. Here Y and X are uniformly integrable càdllàg Hilbert space valued martingales and Y differentially subordinate to X via the square bracket process. We also present a second, very simple proof of the special case Y=X. In this generality, notably including processes with jumps, the special case Y = X addresses a question raised in the late 70s by Bonami--Lépingle.

Various sharp estimates for semi-discrete Riesz transforms of the second order,

preprint, joint with K. Domelevo, A. Osekowski

We give several sharp estimates for a class of combinations of second order Riesz transforms on Lie groups G=Gx×Gy that are multiply connected, composed of a discrete abelian component Gx and a connected component Gy endowed with a biinvariant measure.

Dimensionless Lp estimates for the Riesz vector on manifolds,

preprint, joint with K. Dahmani, K. Domelevo, K. Skreb

We present a new proof of the dimensionless Lp boundedness of the Riesz vector on manifolds with bounded geometry. Our proof has the significant advantage that it allows for a much stronger conclusion, namely that of a new dimensionless weighted Lp estimate with optimal exponent. We use sparse domination with continuous index.

A matrix weighted bilinear Carleson Lemma and Maximal Function,

preprint, joint with S. Pott, M. Reguera

We prove a weak bilinear Carleson Lemma with matrix weight. We also show that a maximal function `with poor memory' has dimensionless estimates.

Failure of the matrix weighted bilinear Carleson embedding theorem,

preprint, joint with K. Domelevo, K. Skreb

We prove that the weak bilinear Carleson Lemma mentioned above is optimal. In other words, any bilinear Carleson Lemma that would be useful for an improvement of matrix weighted estimates fails.