Harmonic Analysis
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Proof of an extension of E. Sawyer's conjecture about weighted mixed weak-type estimates
(Mathematische Annalen, 2018-09)We show that if $v\in A_\infty$ and $u\in A_1$, then there is a constant $c$ depending on the $A_1$ constant of $u$ and the $A_{\infty}$ constant of $v$ such that $$\Big\|\frac{ T(fv)} {v}\Big\|_{L^{1,\infty}(uv)}\le c\, ... -
Weighted norm inequalities for rough singular integral operators
(Journal of Geometric Analysis, 2018-08-17)In this paper we provide weighted estimates for rough operators, including rough homogeneous singular integrals $T_\Omega$ with $\Omega\in L^\infty(\mathbb{S}^{n-1})$ and the Bochner--Riesz multiplier at the critical index ... -
On Bloom type estimates for iterated commutators of fractional integrals
(Indiana University Mathematics Journal, 2018-04)In this paper we provide quantitative Bloom type estimates for iterated commutators of fractional integrals improving and extending results from [15]. We give new proofs for those inequalities relying upon a new sparse ... -
Quantitative weighted estimates for singular integrals and commutators
(2018-02-27)In this dissertation several quantitative weighted estimates for singular integral op- erators, commutators and some vector valued extensions are obtained. In particular strong and weak type $(p, p)$ estimates, Coifman-Fe ... -
Bilinear representation theorem
(Transactions of the American Mathematical Society, 2018-01-01)We represent a general bilinear Calderón--Zygmund operator as a sum of simple dyadic operators. The appearing dyadic operators also admit a simple proof of a sparse bound. In particular, the representation implies a so ... -
Improved A1 − A∞ and related estimates for commutators of rough singular integrals
(Proceedings of the Edinburgh Mathematical Society, 2017)An $A_1-A_\infty$ estimate improving a previous result in [22] for $[b, T_\Omega]$ with $\Omega\in L^\infty(S^{n-1})$ and $b\in BMO$ is obtained. Also a new result in terms of the $A_\infty$ constant and the one ... -
Weighted mixed weak-type inequalities for multilinear operators
(Studia Mathematica, 2017)In this paper we present a theorem that generalizes Sawyer's classic result about mixed weighted inequalities to the multilinear context. Let $\vec{w}=(w_1,...,w_m)$ and $\nu = w_1^\frac{1}{m}...w_m^\frac{1}{m}$, the main ... -
Mixed norm estimates for the Cesàro means associated with Dunkl-Hermite expansions
(Transactions of the American Mathematical Society, 2017)Our main goal in this article is to study mixed norm estimates for the Cesàro means associated with Dunkl--Hermite expansions on $\mathbb{R}^d$. These expansions arise when one considers the Dunkl--Hermite operator ... -
On pointwise and weighted estimates for commutators of Calderón-Zygmund operators
(Advances in Mathematics, 2017)In recent years, it has been well understood that a Calderón-Zygmund operator T is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar ... -
Weak and strong $A_p$-$A_\infty$ estimates for square functions and related operators
(Proceedings of the American Mathematical Society, 2017-07)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 $[w]_{A_p} ... -
Sharp weighted estimates involving one supremum
(Comptes Rendus Mathematique, 2017-07)In this note, we study the sharp weighted estimate involving one supremum. In particular, we give a positive answer to an open question raised by Lerner and Moen. We also extend the result to rough homogeneous singular ... -
A quantitative approach to weighted Carleson condition
(Concrete Operators, 2017-05-05)Quantitative versions of weighted estimates obtained by F. Ruiz and J.L. Torrea for the operator \[ \mathcal{M}f(x,t)=\sup_{x\in Q,\,l(Q)\geq t}\frac{1}{|Q|}\int_{Q}|f(x)|dx \qquad x\in\mathbb{R}^{n}, \, t \geq0 \] are ... -
Sparse domination theorem for multilinear singular integral operators with $L^{r}$-Hörmander condition
(Michigan Mathematical Journal, 2017-04-01)In this note, we show that if $T$ is a multilinear singular integral operator associated with a kernel satisfies the so-called multilinear $L^{r}$-Hörmander condition, then $T$ can be dominated by multilinear sparse operators. -
A characterization of two weight norm inequality for Littlewood-Paley $g_{\lambda}^{*}$-function
(Journal of Geometric Analysis, 2017)Let $n\ge 2$ and $g_{\lambda}^{*}$ be the well-known high dimensional Littlewood-Paley function which was defined and studied by E. M. Stein, $$g_{\lambda}^{*}(f)(x)=\bigg(\iint_{\mathbb R^{n+1}_{+}} \Big(\frac{t}{t+|x-y ... -
Quantitative weighted estimates for rough homogeneous singular integrals
(Israel Journal of Mathematics, 2017-03-11)We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space $L^2(w)$, we obtain a bound ... -
The Calderón problem with corrupted data
(Inverse Problems, 2017-01)We consider the inverse Calderón problem consisting of determining the conductivity inside a medium by electrical measurements on its surface. Ideally, these measurements determine the Dirichlet-to-Neumann map and, therefore, ... -
Inverse scattering for a random potential
(2016-05)In this paper we consider an inverse problem for the $n$-dimensional random Schrödinger equation $(\Delta-q+k^2)u = 0$. We study the scattering of plane waves in the presence of a potential $q$ which is assumed to be a ... -
Global Uniqueness for The Calderón Problem with Lipschitz Conductivities
(Forum of Mathematics, Pi, 2016-01-01)We prove uniqueness for the Calderón problem with Lipschitz conductivities in higher dimensions. Combined with the recent work of Haberman, who treated the three- and four-dimensional cases, this confirms a conjecture of ... -
New bounds for bilinear Calderón-Zygmund operators and applications
(Revista Matemática Iberoamericana, 2016-11-25)In this work we extend Lacey’s domination theorem to prove the pointwise control of bilinear Calderón–Zygmund operators with Dini–continuous kernel by sparse operators. The precise bounds are carefully tracked following ... -
On sums involving Fourier coefficients of Maass forms for SL(3,Z)
(2016-09-10)We derive a truncated Voronoi identity for rationally additively twisted sums of Fourier coefficients of Maass forms for SL(3,Z), and as an application obtain a pointwise estimate and a second moment estimate for the sums ...
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