## About

52

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Education

January 1996 - August 2000

## Publications

Publications (52)

The convex body maximal operator is a natural generalization of the Hardy–Littlewood maximal operator. In this paper we are considering its dyadic version in the presence of a matrix weight. To our surprise it turns out that this operator is not bounded. This is in a sharp contrast to a Doob's inequality in this context. At first, we show that the...

The convex body maximal operator is a natural generalisation of the Hardy Littlewood maximal operator. In the presence of a matrix weight it is not bounded.

Let \((T_t)_{t \geqslant 0}\) be a markovian (resp. submarkovian) semigroup on some \(\sigma \)-finite measure space \((\Omega ,\mu )\). We prove that its negative generator A has a bounded \(H^{\infty }(\Sigma _\theta )\) calculus on the weighted space \(L^2(\Omega ,wd\mu )\) as long as the weight \(w : \Omega \rightarrow (0,\infty )\) has finite...

We develop a biparameter theory for matrix weights and show various bounds for Journ\'{e} operators under the assumption of the matrix Muckenhoupt condition.

We characterize dyadic little BMO via the boundedness of the tensor commutator with a single well chosen dyadic shift. It is shown that several proof strategies work for this problem, both in the unweighted case as well as with Bloom weights. Moreover, we address the flexibility of one of our methods.

Let (Tt) t 0 be a markovian (resp. submarkovian) semigroup on some $\sigma$-finite measure space ($\Omega$, $\mu$). We prove that its negative generator A has a bounded H $\infty$ ($\Sigma$ $\theta$) calculus on the weighted space L 2 ($\Omega$, wd$\mu$) as long as the weight w : $\Omega$ $\rightarrow$ (0, $\infty$) has finite characteristic define...

We prove a bilinear Carleson embedding theorem with matrix weight and scalar measure. In the scalar case, this becomes exactly the well known weighted bilinear Carleson embedding theorem. Although only allowing scalar Carleson measures, it is to date the only extension to the bilinear setting of the recent Carleson embedding theorem by Culiuc and T...

We show that the classical $A_{\infty}$ condition is not sufficient for a lower square function estimate in the non-homogeneous weighted $L^2$ space. We also show that under the martingale $A_2$ condition, an estimate holds true, but the optimal power of the characteristic jumps from $1 / 2$ to $1$ even when considering the classical $A_2$ characte...

We prove failure of the natural formulation of a matrix weighted bilinear Carleson embedding theorem, featuring a matrix–valued Carleson sequence as well as products of norms for the embedding. We show that assuming an A2 weight is also not sufficient. Indeed, a uniform bound on the conditioning number of the matrix weight is necessary and sufficie...

We prove failure of the natural formulation of a matrix weighted bilinear Carleson embedding theorem, featuring a matrix valued Carleson sequence as well as products of norms for the embedding. We show that assuming an A2 weight is also not sufficient. Indeed, a uniform bound on the conditioning number of the matrix weight is necessary and sufficie...

We prove a bilinear Carleson embedding theorem with matrix weight and scalar measure. In the scalar case, this becomes exactly the well known weighted bilinear Carleson embedding theorem. Although only allowing scalar Carleson measures, it is to date the only extension to the bilinear setting of the recent Carleson embedding theorem by Culiuc and T...

We give several sharp estimates for a class of combinations of second-order Riesz transforms on Lie groups G = GxxGy that are multiply connected, composed of a discrete Abelian component Gx and a connected component Gy endowed with a biinvariant measure. These estimates include new sharp Lp estimates via Choi type constants, depending upon the mult...

We present a new proof of the dimensionless $L^p$ 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 $L^p$ estimate with optimal exponent. Other than previous arguments, only a small part of our proof i...

**The published version of this article will be posted soon and the link updated accordingly**
The sharp square function estimate with matrix weight, Discrete Analysis 2019:2, 8 pp.
A central theme in harmonic analysis is to determine whether naturally occurring linear operators are bounded with respect to naturally occurring norms. One class of...

This article extends the recent two-weight theory of Bloom BMO and commutators with Calder\'on-Zygmund operators to the biparameter setting. Specifically, if $\mu$ and $\lambda$ are biparameter $A_p$ weights, $\nu := \mu^{1/p}\lambda^{-1/p}$ is the Bloom weight, and $b$ is in the weighted little bmo space $bmo(\nu)$, then the commutator $[b, T]$ is...

We give several sharp estimates for a class of combinations of second order Riesz transforms on Lie groups ${G}={G}_{x} \times {G}_{y}$ that are multiply connected, composed of a discrete abelian component ${G}_{x}$ and a connected component ${G}_{y}$ endowed with a biinvariant measure. These estimates include new sharp $L^p$ estimates via Choi typ...

We introduce the so called convex body valued sparse operators, which generalize the notion of sparse operators to the case of spaces of vector valued functions. We prove that Calder\'on--Zygmund operators as well as Haar shifts and paraproducts can be dominated by such operators. By estimating sparse operators we obtain weighted estimates with mat...

We study a class of combinations of second order Riesz transforms on Lie groups \(\mathbb {G}=\mathbb {G}_{x} \times \mathbb {G}_{y}\) that are multiply connected, composed of a discrete abelian component \(\mathbb {G}_{x}\) and a compact connected component \(\mathbb {G}_{y}\). We prove sharp L
p
estimates for these operators, therefore generalizi...

We prove a sharp weighted $L^p$ estimate of $Y^{\ast}$ with respect to $X$. Here $Y$ and $X$ are uniformly integrable cadlag Hilbert space valued martingales and $Y$ differentially subordinate to $X$ via the square bracket process. The proof is via an iterated stopping procedure and self similarity argument known as 'sparse domination'. We point ou...

We prove optimal ${L}^2$ bounds for a pair of Hilbert space valued differentially subordinate martingales under a change of law. The change of law is given by a process called a weight and sharpness in this context refers to the optimal growth with respect to the characteristic of the weight. The pair of martingales are adapted, uniformly integrabl...

We characterize Lp boundedness of iterated commutators of multiplication by a symbol function and tensor products of Riesz and Hilbert transforms. We obtain a two-sided norm estimate that shows that such operators are bounded on Lp if and only if the symbol belongs to the appropriate multi-parameter BMO class. We extend our results to a much more i...

We show that the centered discrete Hilbert transform on integers applied to a
function can be written as the conditional expectation of a transform of
stochastic integrals, where the stochastic processes considered have jump
components. The stochastic representation of the function and that of its
Hilbert transform are under differential subordinat...

We show that the norm of the vector of Riesz transforms as operator in the
weighted Lebesgue space L^2(w) is bounded by a constant multiple of the first
power of the Poisson-A_2 characteristic of w. The bound is free of dimension.
Our argument requires an extension of Wittwer's linear estimate for martingale
transforms to the vector valued setting...

We consider iterated commutators of multiplication by a symbol function and
tensor products of Hilbert or Riesz transforms. We establish mixed BMO classes
of symbols that characterize boundedness of these objects in $L^p$. Little BMO
and product BMO, big Hankel operators and iterated commutators are the base
cases of our results. We use operator th...

We study a class of combinations of second order Riesz transforms on Lie groups $G = G_x \times G_y$ that are multiply connected, composed of a discrete abelian component $G_x$ and a compact connected component $G_y$ . We prove sharp $L^p$ estimates for these operators, therefore generalizing previous results [13][4]. The proof uses stochastic inte...

We show that multipliers of second order Riesz transforms on products of
discrete abelian groups enjoy the $L^{p} $ estimate $p^{\ast} -1$, where
$p^{\ast} = \max \{ p,q \}$ and $p$ and $q$ are conjugate exponents. This
estimate is sharp if one considers all multipliers of the form $\sum_i
\sigma_{i} R_{i} R^{\ast}_{i}$ with $| \sigma_{i} | \leqsla...

Let $W$ denote a matrix $A_2$ weight. In this paper, we implement a scalar
argument using the square function to deduce square-function type results for
vector-valued functions in $L^2(\mathbb{R},\mathbb{C}^d)$. These results are
then used to study the boundedness of the Hilbert transform and Haar
multipliers on $L^2(\mathbb{R},\mathbb{C}^d)$. Our...

We consider iterated commutators of multiplication by a symbol function and smooth Calderon-Zygmund operators, described by Fourier multipliers of homogeneity 0. We establish a criterion for a collection of symbols so that the corresponding Calderon-Zygmund operators characterize product BMO by means of iterated commutators. We therefore extend, in...

As a corollary to our main theorem we give a new proof of the result that the
norm of the Hilbert transform on L^2(w) has norm bounded by a the A_2
characteristic of a weight to the first power, a theorem of one of us. This new
proof begins as the prior proofs do, by passing to Haar shifts. Then, we apply
a deep two-weight T1 theorem of Nazarov-Tre...

We study the possible analogous of the Div-Curl Lemma in classical harmonic
analysis and partial differential equations, but from the point of view of the
multi-parameter setting. In this context we see two possible Div-Curl lemmas
that arise. Extensions to differential forms are also given.

As a corollary to our main theorem we give a new proof of the result that the
norm of the Hilbert transform on L^2(w) has norm bounded by a the A_2
characteristic of a weight to the first power, a theorem of one of us. This new
proof begins as the prior proofs do, by passing to Haar shifts. Then, we apply
a deep two-weight T1 theorem of Nazarov-Tre...

We establish new $p$-estimates for the norm of the generalized
Beurling--Ahlfors transform $\mathcal{S}$ acting on form-valued functions.
Namely, we prove that $\norm{\mathcal{S}}_{L^p(\R^n;\Lambda)\to
L^p(\R^n;\Lambda)}\leq n(p^{*}-1)$ where $p^*=\max\{p, p/(p-1)\},$ thus
extending the recent Nazarov--Volberg estimates to higher dimensions. The
ev...

We give a simple proof of L^p boundedness of iterated commutators of Riesz transforms and a product BMO function. We use a representation of the Riesz transforms by means of simple dyadic operators - dyadic shifts - which in turn reduces the estimate quickly to paraproduct estimates. Comment: 10 pages, submitted to Proceedings of El Escorial 2008

We establish sufficient conditions on the two weights w and v so that the Beurling-Ahlfors transform acts continuously from L2(w-1) to L2(v). Our conditions are simple estimates involving heat extensions and Green's potentials of the weights.

We show that the norm of the Hilbert transform as an operator in the weighted space
(ω) for 2 ≤ p p characteristic of ω. This result is sharp. We also prove a bilinear imbedding theorem with simple conditions.

We establish Lp-solvability for 1<p<∞ of the Dirichlet problem on Lipschitz domains with small Lipschitz constants for elliptic divergence and non-divergence type operators with rough coefficients obeying a certain Carleson condition with small norm.

It is shown that product BMO of Chang and Fefferman, defined on the product of Euclidean spaces can be characterized by the multiparameter commutators of Riesz transforms. This extends a classical one-parameter result of Coifman, Rochberg, and Weiss, and at the same time extends the work of Lacey and Ferguson and Lacey and Terwilleger on multiparam...

We establish the best possible bound on the norm of the Riesz transforms as operators in the weighted space LpRn(!) for 1 < p < 1 in terms of the classical Ap characteristic of the weight.

In this note we present a new proof of the Carleson Embedding Theorem on the unit disc and unit ball. The only technical tool used in the proof of this fact is Green's formula. The starting point is that every Carleson measure gives rise to a bounded subharmonic function. Using this function we construct a new related Carleson measure that allows f...

We show that the Luzin area integral or the square function on the unit ball of ℂn
, regarded as an operator in the weighted space L
2(w) has a linear bound in terms of the invariant A
2 characteristic of the weight. We show a dimension-free estimate for the “area-integral” associated with the weighted L
2(w) norm of the square function. We prove t...

In [O. Dragičević, A. Volberg, Sharp estimate of the Ahlfors–Beurling operator via averaging martingale transforms, Michigan Math. J. 51 (2) (2003) 415–435] the Ahlfors–Beurling operator T was represented as an average of two-dimensional martingale transforms. The same result can be proven for powers Tn. Motivated by [T. Iwaniec, G. Martin, Riesz t...

The main aspiration of this note is to construct several different Haar-type systems in euclidean spaces of higher dimensions and prove sharp Lp bounds for the corresponding martingale transforms. In dimension one this was a result of Burkholder. The motivation for working in this direction is the search for Lp estimates of the Ahlfors-Beurling ope...

We obtain sharp weighted Lp estimates in the Rubio de Francia extrapolation the- orem in terms of the Ap characteristic constant of the weight. Precisely, if for a given 1 < r < 1 the norm of a sublinear operator on Lr(w) is bounded by a function of the Ar characteristic constant of the weight w, then for p > r it is bounded on Lp(v) by the same in...

Let W be an operator weight, i.e., a weight function taking values in the bounded linear operators on a Hilbert space ℋ. We prove that if the dyadic martingale transforms are uniformly bounded on L ℝ 2 (W) for each dyadic grid in ℝ, then the Hilbert transform is bounded on L ℝ 2 (W) as well, thus providing an analogue of Burkholder’s theorem for op...

The first author showed in [18] that the Hilbert transform lies in the closed convex hull of dyadic singular operators - so called dyadic shifts. We show here that the same is true in any Rn - the Riesz transforms can be obtained as the results of averaging of dyadic shifts. The goal of this paper is almost entirely methodological: we simplify the...

We establish borderline regularity for solutions of the Beltrami equation $f\sb z-\mu f\sb {\overline {z}}=0$ on the plane, where $\mu$ is a bounded measurable function, $\parallel\mu\parallel\sb \infty=k<1$. What is the minimal requirement of the type $f\in W \sp {1,q}\sb {{\rm loc}}$ which guarantees that any solution of the Beltrami equation wit...

We show that the norm of the Hilbert transform as an operator on the weighted space L2(ω) is bounded by a constant multiple of the 3/2 power of the A2 constant of w, in other words by c supI(〈ω〉I〈ω-1〉 I)3/2. We also give a short proof for sharp upper and lower bounds for the dyadic square function.

We are going to show that the commutator of the Hilbert transform with matrix multiplication by a BMO matrix of size n×n is bounded by a multiple of logn times the BMO-norm of the matrix.RésuméNous allons démontrer une majoration de la forme pour une fonction matricielle B de BMO.

We show that the norm of the Riesz transforms as operators in the weighted Lebesgue space L2! are bounded by a constant multiple of the first power of the Poisson A2 characteristic of !. The bound is free of dimension. For n > 1 the Poisson A2 class and the classical A2 class are not the same.