arXiv subjects

Ezequiel Rela

Publications and source records attributed to Ezequiel Rela.

25 records · Page 2Linked to original sources

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

Small Furstenberg sets

For $α$ in $(0,1]$, a subset $E$ of $\RR$ is called Furstenberg set of type $α$ or $F_α$-set if for each direction $e$ in the unit circle there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap\ell_e$ is greater or equal than $α$. In this paper we show that if $α> 0$, there exists a set $E\in F_α$ such that $\HH{g}(E)=0$ for $g(x)=x^{1/2+3/2α}\log^{-θ}(\frac{1}{x})$, $θ>\frac{1+3α}{2}$, which improves on the the previously known bound, that $H^β(E) = 0$ for $β>1/2+3/2α$. Further, by refining the argument in a subtle way, we are able to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for $\h_γ(x)=\log^{-γ}(\frac{1}{x})$, $γ>0$, we construct a set $E_γ\in F_{\h_γ}$ of Hausdorff dimension not greater than 1/2. Since in a previous work we showed that 1/2 is a lower bound for the Hausdorff dimension of any $E\in F_{\h_γ}$, with the present construction, the value 1/2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions $\h_γ$.

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↗

arXiv2010

Furstenberg sets for a fractal set of directions

In this note we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair $α,β\in(0,1]$, we will say that a set $E\subset \R^2$ is an $F_{αβ}$-set if there is a subset $L$ of the unit circle of Hausdorff dimension at least $β$ and, for each direction $e$ in $L$, there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap\ell_e$ is equal or greater than $α$. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that $\dim(E)\ge\max\left\{α+\fracβ{2} ; 2α+β-1\right\}$ for any $E\in F_{αβ}$. In particular we are able to extend previously known results to the ``endpoint'' $α=0$ case.

math.CA↗

arXiv2009

Improving dimension estimates for Furstenberg-type sets

In this paper we prove some lower bounds on the Hausdorff dimension of sets of Furstenberg type. Moreover, we extend these results to sets of generalized Furstenberg type, associated to doubling dimension functions. With some additional growth conditions on the dimension function, we obtain a lower bound on the dimension of "zero dimensional" Furstenberg sets.

math.CA↗