arXiv subjects

Carlos Pérez

Publications and source records attributed to Carlos Pérez.

At least 37 records · Page 2Linked to original sources

arXiv2019

Extensions of the John-Nirenberg theorem and applications

The John-Nirenberg theorem states that functions of bounded mean oscillation are exponentially integrable. In this article we give two extensions of this theorem. The first one relates the dyadic maximal function to the sharp maximal function of Fefferman-Stein, while the second one concerns local weighted mean oscillations, generalizing a result of Muckenhoupt and Wheeden. Applications to the context of generalized Poincaré type inequalities and to the context of the $C_p$ class of weights are given. Extensions to the case of polynomial BMO type spaces are also given.

math.CA↗

arXiv2019

A note on generalized Fujii-Wilson conditions and BMO spaces

In this note we generalize the definition of Fujii-Wilson condition providing quantitative characterizations of some interesting classes of weights, such as $A_\infty$, $A_\infty^{weak}$ and $C_p$, in terms of BMO type spaces suited to them. We will provide as well some self improvement properties for some of those generalized BMO spaces and some quantitative estimates for Bloom's BMO type spaces.

math.CA↗

arXiv2019

Degenerate Poincaré-Sobolev inequalities

We study weighted Poincaré and Poincaré-Sobolev type inequalities with an explicit analysis on the dependence on the $A_p$ constants of the involved weights. We obtain inequalities of the form $$ \left (\frac{1}{w(Q)}\int_Q|f-f_Q|^{q}w\right )^\frac{1}{q}\le C_w\ell(Q)\left (\frac{1}{w(Q)}\int_Q |\nabla f|^p w\right )^\frac{1}{p}, $$ with different quantitative estimates for both the exponent $q$ and the constant $C_w$. We will derive those estimates together with a large variety of related results as a consequence of a general selfimproving property shared by functions satisfying the inequality $$ \frac{1}{|Q|}\int_Q |f-f_Q| dμ\le a(Q), $$ for all cubes $Q\subset\mathbb{R}^n$ and where $a$ is some functional that obeys a specific discrete geometrical summability condition. We introduce a Sobolev-type exponent $p^*_w>p$ associated to the weight $w$ and obtain further improvements involving $L^{p^*_w}$ norms on the left hand side of the inequality above. For the endpoint case of $A_1$ weights we reach the classical critical Sobolev exponent $p^*=\frac{pn}{n-p}$ which is the largest possible and provide different type of quantitative estimates for $C_w$. We also show that this best possible estimate cannot hold with an exponent on the $A_1$ constant smaller than $1/p$. We also provide an argument based on extrapolation ideas showing that there is no $(p,p)$, $p\geq1$, Poincaré inequality valid for the whole class of $RH_\infty$ weights by showing their intimate connection with the failure of Poincaré inequalities, $(p,p)$ in the range $0<p<1$.

math.CA↗

arXiv2018

Regularity of maximal functions on Hardy-Sobolev spaces

We prove that maximal operators of convolution type associated to smooth kernels are bounded in the homogeneous Hardy-Sobolev spaces $\dot{H}^{1,p}(\mathbb{R}^d)$ when $1/p < 1+1/d$. This range of exponents is sharp. As a by-product of the proof, we obtain similar results for the local Hardy-Sobolev spaces $\dot{h}^{1,p}(\mathbb{R}^d)$ in the same range of exponents.

math.CA↗

arXiv2018

Weighted norm inequalities for rough singular integral operators

In this paper we provide weighted estimates for rough operators, including rough homogeneous singular integrals $T_Ω$ with $Ω\in L^\infty(\mathbb{S}^{n-1})$ 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 $Ω\in L^q(\mathbb{S}^{n-1})$, $1<q<\infty$, 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. [CACDPO], results by the first author in [L], suitable adaptations of Rubio de Francia algorithm, the extrapolation theorems for $A_{\infty}$ weights [CMP,CGMP] and ideas contained in previous works by A. Seeger in [S] and D. Fan and S. Sato [FS].

math.CA↗

arXiv2017

A lower bound for $A_p$ exponents for some weighted weak-type inequalities

We give a weak-type counterpart of the main result in an earlier work of the first author, E. Rela and T. Luque which allows to provide a lower bound for the exponent of the $A_{p}$ constant in terms of the behaviour of the unweighted inequalities when $p\rightarrow\infty$ and when $p\rightarrow1^{+}$. We also provide some applications to classical operators.

math.CA↗

arXiv2017

Proof of an extension of E. Sawyer's conjecture about weighted mixed weak-type estimates

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 $$\Big\|\frac{ T(fv)} {v}\Big\|_{L^{1,\infty}(uv)}\le c\, \|f\|_{L^1(uv)},$$ where $T$ can be the Hardy-Littlewood maximal function or any Calderón-Zygmund operator. This result was conjectured in [IMRN, (30)2005, 1849--1871] and constitutes the most singular case of some extensions of several problems proposed by E. Sawyer and Muckenhoupt and Wheeden. We also improve and extends several quantitative estimates.

math.CA↗

arXiv2017

Three observations on commutators of Singular Integral Operators with BMO functions

This paper contains three observations on commutators of Singular Integral Operators with BMO functions: 1) The subgaussian local decay for the commutator, namely \[\frac{1}{|Q|}\left|\left\{x\in Q\, : \, |[b,T](fχ_Q)(x)|>M^2f(x)t\right\}\right|\leq c e^{-\sqrt{ct\|b\|_{BMO}}} \] is sharp, that is, it is subgaussian and not better. 2) It is not possible to obtain a pointwise control of the commutator by a finite sum of sparse operators defined with $L\log L$ averages. 3) If $w\in A_p\setminus A_1$ then $\left\| wM\left(\frac{f}{w}\right)\right\|_{L^1(\mathbb{R}^n)\rightarrow L^{1,\infty}(\mathbb{R}^n)}=\infty$.

math.CA↗

arXiv2016

Equilibrium Graphs

In this paper we present an extension of Peirce's existential graphs to provide a diagrammatic representation of expressions in Quantified Equilibrium Logic (QEL). Using this formalisation, logical connectives are replaced by encircled regions (circles and squares) and quantified variables are represented as "identity" lines. Although the expressive power is equivalent to that of QEL, the new representation can be useful for illustrative or educational purposes.

cs.AI↗

arXiv2016

Borderline weighted estimates for commutators of singular integrals

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

math.CA↗

arXiv2015

Reverse Hölder Property for strong weights and general measures

We present dimension-free reverse Hölder inequalities for strong $A^*_p$ weights, $1\le p < \infty$. We also provide a proof for the full range of local integrability of $A_1^*$ weights. The common ingredient is a multidimensional version of Riesz's "rising sun" lemma. Our results are valid for any nonnegative Radon measure with no atoms. For $p=\infty$, we also provide a reverse Hölder inequality for certain product measures. As a corollary we derive mixed $A_p^*-A_\infty^*$ weighted estimates.

math.CA↗

arXiv2015

The $L(\log L)^ε$ endpoint estimate for maximal singular integral operators

We prove in this paper the following estimate for the maximal operator $T^*$ associated to the singular integral operator $T$: $ \|T^*f\|_{L^{1,\infty}(w)} \lesssim \frac{1}ε \int_{\mathbb{R}^n} |f(x)| M_{L(\log L)^ε} (w)(x)dx$, for $w\geq 0, 0<ε\leq 1.$ This follows from the sharp $L^p$ estimate $ \|T^*f \|_{ L^{p}(w) } \lesssim p' (\frac{1}δ)^{1/p'} \|f \|_{L^{p}(M_{ L(\log L)^{p-1+δ}} (w))}$, for $1<p<\infty, w\geq 0, 0<δ\leq 1. $ As as a consequence we deduce that $ \|T^*f\|_{L^{1,\infty}(w)} \lesssim [w]_{A_{1}} \log(e+ [w]_{A_{\infty}}) \int_{\mathbb{R}^n} |f| w dx, $ extending the endpoint results obtained in [LOP] and [HP] to maximal singular integrals. Another consequence is a quantitative two weight bump estimate.

math.CA↗

arXiv2013

Optimal exponents in weighted estimates without examples

We present a general approach for proving the optimality of the exponents on weighted estimates. We show that if an operator $T$ satisfies a bound like $$ \|T\|_{L^{p}(w)}\le c\, [w]^β_{A_p} \qquad w \in A_{p}, $$ then the optimal lower bound for $β$ is closely related to the asymptotic behaviour of the unweighted $L^p$ norm $\|T\|_{L^p(\mathbb{R}^n)}$ as $p$ goes to 1 and $+\infty$, which is related to Yano's classical extrapolation theorem. By combining these results with the known weighted inequalities, we derive the sharpness of the exponents, without building any specific example, for a wide class of operators including maximal-type, Calderón--Zygmund and fractional operators. In particular, we obtain a lower bound for the best possible exponent for Bochner-Riesz multipliers. We also present a new result concerning a continuum family of maximal operators on the scale of logarithmic Orlicz functions. Further, our method allows to consider in a unified way maximal operators defined over very general Muckenhoupt bases.

math.CA↗

arXiv2013

A new quantitative two weight theorem for the Hardy-Littlewood maximal operator

A quantitative two weight theorem for the Hardy-Littlewood maximal operator is proved improving the known ones. As a consequence a new proof of the main results in [HP] and in [HPR12] is obtained which avoids the use of the sharp quantitative reverse Holder inequality for $A_{\infty}$ proved in those papers. Our results are valid within the context of spaces of homogeneous type without imposing the non-empty annuli condition.

math.CA↗

arXiv2013

Exponential decay estimates for Singular Integral operators

The following subexponential estimate for commutators is proved |[|\{x\in Q: |[b,T]f(x)|>tM^2f(x)\}|\leq c\,e^{-\sqrt{α\, t\|b\|_{BMO}}}\, |Q|, \qquad t>0.\] where $c$ and $α$ are absolute constants, $T$ is a Calderón--Zygmund operator, $M$ is the Hardy Littlewood maximal function and $f$ is any function supported on the cube $Q$. It is also obtained \[|\{x\in Q: |f(x)-m_f(Q)|>tM_{1/4;Q}^#(f)(x) \}|\le c\, e^{-α\,t}|Q|,\qquad t>0,\] where $m_f(Q)$ is the median value of $f$ on the cube $Q$ and $M_{1/4;Q}^#$ is Strömberg's local sharp maximal function. As a consequence it is derived Karagulyan's estimate \[|\{x\in Q: |Tf(x)|> tMf(x)\}|\le c\, e^{-c\, t}\,|Q|\qquad t>0,\] improving Buckley's theorem. A completely different approach is used based on a combination of "Lerner's formula" with some special weighted estimates of Coifman-Fefferman obtained via Rubio de Francia's algorithm. The method is flexible enough to derive similar estimates for other operators such as multilinear Calderón--Zygmund operators, dyadic and continuous square functions and vector valued extensions of both maximal functions and Calderón--Zygmund operators. On each case, $M$ will be replaced by a suitable maximal operator.

math.CA↗

arXiv2012

Sharp weighted bounds for multilinear maximal functions and Calderón-Zygmund operators

In this paper we prove some sharp weighted norm inequalities for the multi(sub)linear maximal function $\Mm$ introduced in \cite{LOPTT} and for multilinear Calderón-Zygmund operators. In particular we obtain a sharp mixed "$A_p-A_{\infty}$" bound for $\Mm$, some partial results related to a Buckley-type estimate for $\Mm$, and a sufficient condition for the boundedness of $\Mm$ between weighted $L^p$ spaces with different weights taking into account the precise bounds. Next we get a bound for multilinear Calderón-Zygmund operators in terms of dyadic positive multilinear operators in the spirit of the recent work. Then we obtain a multilinear version of the "$A_2$ conjecture". Several open problems are posed.

math.CA↗

arXiv2012

Sharp Reverse Hölder property for A_\infty weights on spaces of homogeneous type

In this article we present a new proof of a sharp Reverse Hölder Inequality for $A_\infty$ weights that is valid in the context of spaces of homogeneous type. Then we derive two applications: a precise open property of Muckenhoupt classes and, as a consequence of this last result, we obtain a simple proof of a sharp weighted bound for the Hardy-Littlewood maximal function involving $A_\infty$ constants: |M|_{L^p(w)} \leq c (\frac{1}{p-1} [w]_{A_p}[σ]_{A_\infty})^{1/p}, where $1<p<\infty$, $σ=w^{\frac{1}{1-p}}$ and $c$ depends only on the doubling constant of the measure $μ$ and the geometric constant $κ$ of the quasimetric.

math.CA↗

arXiv2012

Improving bounds for singular operators via Sharp Reverse Hölder Inequality for $A_{\infty}$

In this expository article we collect and discuss some recent results on different consequences of a Sharp Reverse Hölder Inequality for $A_{\infty}$ weights. For two given operators $T$ and $S$, we study $L^p(w)$ bounds of Coifman-Fefferman type. $ \|Tf\|_{L^p(w)}\le c_{n,w,p} \|Sf\|_{L^p(w)}, $ that can be understood as a way to control $T$ by $S$. We will focus on a \emph{quantitative} analysis of the constants involved and show that we can improve classical results regarding the dependence on the weight $w$ in terms of Wilson's $A_{\infty}$ constant $ [w]_{A_{\infty}}:=\sup_Q\frac{1}{w(Q)}\int_Q M(wχ_Q).$ We will also exhibit recent improvements on the problem of finding sharp constants for weighted norm inequalities involving several singular operators. We obtain mixed $A_{1}$--$A_{\infty}$ estimates for the commutator $[b,T]$ and for its higher order analogue $T^k_{b}$. A common ingredient in the proofs presented here is a recent improvement of the Reverse Hölder Inequality for $A_{\infty}$ weights involving Wilson's constant.

math.CA↗