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Carlos Perez

Publications and source records attributed to Carlos Perez.

13 recordsLinked to original sources
arXiv2020

Minimal conditions for BMO

We study minimal integrability conditions via Luxemburg-type expressions with respect to generalized oscillations that imply the membership of a given function $f$ to the space BMO. Our method is simple, sharp and flexible enough to be adapted to several different settings, like spaces of homogeneous type, non doubling measures on $\mathbb{R}^n$ and also BMO spaces defined over more general bases than the basis of cubes.

math.CA↗
arXiv2017

Vector-valued operators, optimal weighted estimates and the $C_p$ condition

Sharp weighted estimates are obtained for vector-valued extensions of the Hardy-Littlewood maximal operator, Calderón-Zygmund operators and Coifman-Rochberg-Weiss commutator. Those estimates will rely upon suitable pointwise estimates in terms of sparse operators. We also prove some new results for the $C_p$ classes introduced by Muckenhoupt and later extended by Sawyer, in particular we extend the result to the full expected range $p > 0$, to the weak norm, to other operators and to the their vector-valued extensions.

math.CA↗
arXiv2015

Quantitative weighted mixed weak-type inequalities for classical operators

We improve on several mixed weak type inequalities both for the Hardy-Littlewood maximal function and for Calderón-Zygmund operators. These type of inequalities were considered by Muckenhoupt and Wheeden and later on by Sawyer estimating the $L^{1, \infty}(uv)$ norm of $v^{-1}T(fv)$ for special cases. The emphasis is made in proving new and more precise quantitative estimates involving the $A_p$ or $A_\infty$ constants of the weights involved.

math.CA↗
arXiv2011

Sharp weighted estimates for classical operators

We give a new proof of the sharp one weight $L^p$ inequality for any operator $T$ that can be approximated by Haar shift operators such as the Hilbert transform, any Riesz transform, the Beurling-Ahlfors operator. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators. Our method is flexible enough to prove the corresponding sharp one-weight norm inequalities for some operators of harmonic analysis: the maximal singular integrals associated to $T$, Dyadic square functions and paraproducts, and the vector-valued maximal operator of C. Fefferman-Stein. Also we can derive a very sharp two-weight bump type condition for $T$.

math.CA↗
arXiv2011

The multilinear strong maximal function

A multivariable version of the strong maximal function is introduced and a sharp distributional estimate for this operator in the spirit of the Jessen, Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize the boundedness of this multivariable operator on products of weighted Lebesgue spaces equipped with multiple weights are obtained. Results for other multi(sub)linear maximal functions associated with bases of open sets are studied too. Certain bilinear interpolation results between distributional estimates, such as that obtained for the multivariable strong maximal function, are also proved.

math.CA↗
arXiv2011

End-point estimates for iterated commutators of multilinear singular integrals

Iterated commutators of multilinear Calderon-Zygmund operators and pointwise multiplication with functions in $BMO$ are studied in products of Lebesgue spaces. Both strong type and weak end-point estimates are obtained, including weighted results involving the vectors weights of the multilinear Calderon-Zygmund theory recently introduced in the literature. Some better than expected estimates for certain multilinear operators are presented too.

math.CA↗
arXiv2010

On $A_2$ conjecture and corona decomposition of weights

We consider here a problem of finding the sharp estimate for the boundedness of an arbitrary Calderón-Zygmund operator in $L^2(w)$, $w\in A_2$. We first prove that for $A_2$ weight $w$ one has that the norm a Calderon--Zygmund operator $T$ in $L^2(w)$ is bounded by the sum of its weak norm, the weak norm of its adjoint, and the $A_2$ norm of the weight. From this result we derive that $\|T\|_{L^2(w)\rightarrow L^2(w)} \le C\,[w]_{A_2}\log (1+[w]_{A_2})$. We believe that the logarithmic factor is superflous. The approach is based on $2$-weight estimates technique and, hence, on non-homogeneous harmonic analysis.

math.AP↗
arXiv2010

Sharp weighted estimates for approximating dyadic operators

We give a new proof of the sharp weighted $L^2$ inequality ||T||_{L^2(w)} \leq c [w]_{A_2} where $T$ is the Hilbert transform, a Riesz transform, the Beurling-Ahlfors operator or any operator that can be approximated by Haar shift operators. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators.

math.CA↗
arXiv2010

Sharp bounds for general commutators on weighted Lebesgue spaces

We show that if an operator T is bounded on weighted Lebesgue space L^2(w) and obeys a linear bound with respect to the A_2 constant of the weight, then its commutator [b,T] with a function b in BMO will obey a quadratic bound with respect to the A_2 constant of the weight. We also prove that the kth-order commutator T^k_b=[b,T^{k-1}_b] will obey a bound that is a power (k+1) of the A_2 constant of the weight. Sharp extrapolation provides corresponding L^p(w) estimates. The results are sharp in terms of the growth of the operator norm with respect to the A_p constant of the weight for all 1<p<\infty, all k, and all dimensions, as examples involving the Riesz transforms, power functions and power weights show.

math.CA↗
arXiv2010

New estimates for the maximal singular integral

In this paper we pursue the study of the problem of controlling the maximal singular integral $T^{*}f$ by the singular integral $Tf$. Here $T$ is a smooth homogeneous Calderón-Zygmund singular integral of convolution type. We consider two forms of control, namely, in the $L^2(\Rn)$ norm and via pointwise estimates of $T^{*}f$ by $M(Tf)$ or $M^2(Tf)$, where $M$ is the Hardy-Littlewood maximal operator and $M^2=M \circ M$ its iteration. It is known that the parity of the kernel plays an essential role in this question. In a previous article we considered the case of even kernels and here we deal with the odd case. Along the way, the question of estimating composition operators of the type $T^\star \circ T$ arises. It turns out that, again, there is a remarkable difference between even and odd kernels. For even kernels we obtain, quite unexpectedly, weak $(1,1)$ estimates, which are no longer true for odd kernels. For odd kernels we obtain sharp weaker inequalities involving a weak $L^1$ estimate for functions in $L LogL$.

math.CA↗
arXiv2010

Sharp weighted bounds for fractional integral operators

The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp boundsare obtained for both the fractional integral operators and the associated fractional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal version of the extrapolation theorem of Rubio de Francia and characterizations of two-weight norm inequalities.

math.CA↗