Improved A1 − A∞ and related estimates for commutators of rough singular integrals

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 supremum $A-A_\infty^{exp}$ constant is proved, providing a counterpart for commutators of the result obained in [19]. Both of the preceding results rely upon a sparse domination result in terms of bilinear forms which is established using techniques from [13]

Quantitative weighted estimates for singular integrals and commutators

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 erman estimates, Fe erman-Stein estimates, Bloom type estimates and endpoint estimates are provided. Most of the proofs of those results rely upon suitable sparse domination results that are provided as well in this dissertation. Also, as an application of the sparse estimates, local ex- ponential decay estimates are revisited, providing new proofs and results for vector valued extensions.BES-2013-06401

A quantitative approach to weighted Carleson condition

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 obtained. As a consequence, some sufficient conditions for the boundedness of $\mathcal{M}$ in the two weight setting in the spirit of the results obtained by C. Pérez and E. Rela and very recently by M. Lacey and S. Spencer for the Hardy-Littlewood maximal operator are derived. As a byproduct some new quantitative estimates for the Poisson integral are obtained

Three Observations on Commutators of Singular Integral Operators with BMO Functions

Three observations on commutators of Singular Integral Operators with BMO functions are exposed, namely 1 - The already known subgaussian local decay for the commutator, namely \[\frac{1}{|Q|}\left|\left\{x\in Q\, : \, |[b,T](f\chi_Q)(x)|>M^2f(x)t\right\}\right|\leq c e^{-\sqrt{ct\|b\|_{BMO}}}\] is sharp, since it cannot be better than subgaussian. 2 - It is not possible to obtain a pointwise control of the commutator by a finite sum of sparse operators defined by $L\log L$ averages. 3 - Motivated by the conjugation method for commutators, it is shown the failure of the following endpoint estimate, if $w\in A_p\setminus A_1$ then $$\left\| wM\left(\frac{f}{w}\right)\right\|_{L^1(\mathbb{R}^n)\rightarrow L^{1,\infty}(\mathbb{R}^n)}=\infty.$$MTM2012-3074

Borderline Weighted Estimates for Commutators of Singular Integrals

In this paper we establish the following estimate $$w\left(\left\{ x\in\mathbb{R}^{n}\,:\,\left|[b,T]f(x)\right| > \lambda\right\} \right)\leq \frac{c_{T}}{\varepsilon^{2}}\int_{\mathbb{R}^{n}}\Phi\left(\|b\|_{BMO}\frac{|f(x)|}{\lambda}\right)M_{L(\log L)^{1+\varepsilon}}w(x)dx$$ where $w\geq0, \, 0<\varepsilon<1$ and $\Phi(t)=t(1+\log^+(t))$. This inequality relies upon the following sharp $L^p$ estimate $$\|[b,T]f\|_{L^{p}(w)}\leq c_{T}\left(p'\right)^{2}p^{2}\left(\frac{p-1}{\delta}\right)^{\frac{1}{p'}} \|b\|_{BMO} \, \|f \|_{L^{p}(M_{L(\log L)^{2p-1+\delta}}w)}$$where $1<p<\infty, w\geq0 \text{ and } 0<\delta<1.$ As a consequence we recover the following estimate essentially contained in \cite{MR3008263}: $$w\left(\{x\in\mathbb{R}^{n}\,:\,\left|[b,T]f(x)\right| >\lambda\}\right)\leq c_T\,[w]_{A_{\infty}}\left(1+\log^{+}[w]_{A_{\infty}}\right)^{2}\int_{\mathbb{R}^{n}} \Phi\left(\|b\|_{BMO}\frac{|f(x)|}{\lambda}\right)Mw(x)dx$$ We also obtain the analogue estimates for symbol-multilinear commutators for a wider class of symbols

Sparse and weighted estimates for generalized Hörmander operators and commutators

In this paper a pointwise sparse domination for generalized Ho ̈rmander and also for iterated commutators with those operators is provided generalizing the sparse domination result in [24]. Relying upon that sparse domination a number of quantitative estimates are derived. Some of them are improvements and complementary results to those contained in a series of papers due to M. Lorente, J. M. Martell, C. P ́erez, S. Riveros and A. de la Torre [29, 28, 27]. Also the quantitative endpoint estimates in [24] are extended to iterated commutators. Other results that are obtained in this work are some local exponential decay estimates for generalized Ho ̈rmander operators in the spirit of [33] and some negative results concerning Coifman-Fefferman estimates for a certain class of kernels satisfying particular generalized Ho ̈rmander conditions

A1A_1 theory of weights for rough homogeneous singular integrals and commutators

Quantitative $A_1-A_\infty$ estimates for rough homogeneous singular integrals $T_{\Omega}$ and commutators of \BMO symbols and $T_{\Omega}$ are obtained. In particular the following estimates are proved: $$\|T_\Omega \|_{L^p(w)}\le c_{n,p}\|\Omega\|_{L^\infty} [w]_{A_1}^{\frac{1}{p}}\,[w]_{A_{\infty}}^{1+\frac{1}{p'}}\|f\|_{L^p(w)}$$ and \| [b,T_{\Omega}]f\| _{L^{p}(w)}\leq c_{n,p}\|b\|_{\BMO}\|\Omega\|_{L^{\infty}} [w]_{A_1}^{\frac{1}{p}}[w]_{A_{\infty}}^{2+\frac{1}{p'}}\|f\|_{L^{p}\left(w\right)}, for $1<p<\infty$ and $1/p+1/p'=1$.BERC 2014-2017 BCAM Severo Ochoa excellence accreditation SEV-2013-0323 MTM2014-53850-P. MTM2015-65888-C04-4-P 2017 Leonardo grant for Researchers and Cultural Creators, BBVA Foundatio

On Bloom type estimates for iterated commutators of fractional integrals

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 domination that we provide as well in this paper and also in techniques de- veloped in the recent paper [22]. We extend as well the necessity established in [15] to iterated commutators providing a new proof. As a consequence of the preceding results we recover the one weight estimates in [7, 1] and es- tablish the sharpness in the iterated case. Our result provides as well a new characterization of the BMO space

Weighted norm inequalities for rough singular integral operators

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 $B_{(n-1)/2}$. More precisely, we prove qualitative and quantitative versions of Coifman--Fefferman type inequalities and their vector-valued extensions, weighted $A_p-A_\infty$ strong and weak type inequalities for $1<p<\infty$, and $A_1-A_\infty$ type weak $(1,1)$ estimates. Moreover, Fefferman--Stein type inequalities are obtained, proving in this way a conjecture raised by the second-named author in the 90's. As a corollary, we obtain the weighted $A_1-A_\infty$ type estimates. Finally, we study rough homogenous singular integrals with a kernel involving a function $\Omega\in L^q(\mathbb{S}^{n-1})$, $1<q<\infty$, and provide Fefferman--Stein inequalities too. The arguments used for our proofs combine several tools: a recent sparse domination result by Conde--Alonso et al. \cite{CACDPO}, results by the first author in \cite{L}, suitable adaptations of Rubio de Francia algorithm, the extrapolation theorems for $A_{\infty}$ weights \cite{CMP,CGMP} and ideas contained in previous works by A. Seeger in \cite{S} and D. Fan and S. Sato \cite{FS}.Juan de la Cierva - Formaci\'on 2015 FJCI-2015-24547. BBVA Foundatio

Vector-valued operators, optimal weighted estimates and the CpC_p condition

In this paper some new results concerning the $C_p$ classes introduced by Muckenhoupt and later extended by Sawyer, are provided. In particular we extend the result to the full range expected $p>0$, to the weak norm, to other operators and to their vector-valued extensions. Some of those results rely upon sparse domination results that in some cases we provide as well. We will also provide sharp weighted estimates for vector valued extensions relying on those sparse domination results.UNLP 11/X752 and PICT 2014-1771 ANPCYT, Argentina. Juan de la Cierva - Formaci\'on 2015 FJCI-2015-24547. CONICET PIP 11220130100329CO, Argentina