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+34 946 567 842
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+34 946 567 842
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cperez@bcamath.org
Information of interest
My research is mainly focused in the area of Real and Harmonic Analysis. More recently I have been interested in the deep connection between the operator norm of some of the basic operators from Harmonic Analysis such as the CalderónZygmund operators, its commutators or maximal functions with the growth of the Ap constants in different various natural spaces. I am also particularly interested in other deep aspects of the theory such as the extrapolation theory and the Reverse Hölder property of the Ap class of weights. I also consider multilinear aspects of the CalderónZygmund theory where many questions are still open. Another good portion of my research interest is related to the theory of multiparameter Harmonic Analysis and its interplay with the theory of weights.
I also like to explore the connection with questions coming from the theory of Schrödinger operators and (degenerate) PoincaréSobolev inequalities.

WEIGHTED FRACTIONAL POINCARE ́ INEQUALITIES VIA ISOPERIMETRIC INEQUALITIES
(2024)Our main result is a weighted fractional Poincar ́e–Sobolev inequality improving the celebrated estimate by Bourgain–Brezis–Mironescu. This also yields an improvement of the classical Meyers–Ziemer theorem in several ways. ...

A TRIBUTE TO POLA HARBOURE: ISOPERIMETRIC INEQUALITIES AND THE HMS EXTRAPOLATION THEOREM
(2023)We give a different simpler proof of the Gagliardo estimate with a measure obtained in [FPW00] and improved in [PR19]. This result will be further improved using fractional Poincaré type inequalities with the extra bonus ...

Remarks on vectorvalued Gagliardo and Poincar ́eSobolev type inequalities with weights
(2023)In this paper we review certain extensions of the Gagliardo and Poincar ́eSobolev type inequalities to later explore the possibility of extending them to the vectorvalued setting. We restrict ourselves to the most classical ...

AN HOMAGE TO GUIDO WEISS AND HIS LEADERSHIP OF THE SAINT LOUIS TEAM: COMMUTATORS OF SINGULAR INTEGRALS AND SOBOLEV INEQUALITIES
(2023)We extend some classical Sobolev type inequalities for linear and nonlinear commutators.

WEIGHTED LORENTZ SPACES: SHARP MIXED Ap − A∞ ESTIMATE FOR MAXIMAL FUNCTIONS
(2023)We prove the sharp mixed Ap − A∞ weighted estimate for the HardyLittlewood maximal function in the context of weighted Lorentz spaces, namely 11 ∥M∥ p,q ≲p,q,n [w]p [σ]min(p,q) , L (w) Ap A∞ 1 where σ = w 1−p . Our ...

ON THE WEIGHTED INEQUALITY BETWEEN THE GAGLIARDO AND SOBOLEV SEMINORMS
(2023)We prove weighted inequalities between the Gagliardo and Sobolev seminorms. With A1 weights we improve earlier results of Bourgain, Brezis, and Mironescu.

POINTWISE ESTIMATES FOR ROUGH OPERATORS WITH APPLICATIONS TO SOBOLEV INEQUALITIES
(2023)We investigate weighted Sobolev inequalities for rough operators. We prove that several operators satisfy a pointwise bound by the Riesz potential applied to the gradient. From this inequality, we derive several new Sobolev ...

Sawyertype inequalities for Lorentz spaces
(202206)The HardyLittlewood maximal operator M satisfies the classical Sawyertype estimate ∥Mfv∥L1,∞(uv)≤Cu,v‖f‖L1(u),where u∈ A1 and uv∈ A∞. We prove a novel extension of this result to the general restricted weak type case. ...

On the BBMPhenomenon in Fractional Poincaré–Sobolev Inequalities with Weights
(2022)In this paper, we unify and improve some of the results of Bourgain, Brezis, and Mironescu and the weighted Poincaré–Sobolev estimate by Fabes, Kenig, and Serapioni. More precisely, we get weighted counterparts of the ...

Some Nonstandard Biparametric Poincaré Type Inequalities Through Harmonic Analysis
(2022)We show some nonstandard Poincaré type estimates in the biparametric setting with appropriate weights. We will derive these results using variants from classical estimates exploiting the interplay between maximal functions ...

Minimal conditions for BMO
(2022)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 ...

Degenerate PoincareSobolev inequalities
(2021)Abstract. We study weighted Poincar ́e and Poincar ́eSobolev type in equalities with an explicit analysis on the dependence on the Ap con stants of the involved weights. We obtain inequalities of the form with different ...

Extensions of the JohnNirenberg theorem and applications
(2021)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 ...

SELFIMPROVING POINCARE ́SOBOLEV TYPE FUNCTIONALS IN PRODUCT SPACES
(2021)In this paper we give a geometric condition which ensures that (q,p)Poincar ́eSobolev inequalities are implied from generalized (1, 1)Poincar ́e inequalities related to L1 norms in the context of product spaces. The ...

A note on generalized FujiiWilson conditions and BMO spaces
(20200701)In this note we generalize the definition of the FujiiWilson condition providing quantitative characterizations of some interesting classes of weights, such as A∞, A∞weak and Cp, in terms of BMO type spaces suited to them. ...

$A_1$ theory of weights for rough homogeneous singular integrals and commutators
(2019)Quantitative $A_1A_\infty$ 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 ...

Regularity of maximal functions on Hardy–Sobolev spaces
(20181201)We prove that maximal operators of convolution type associated to smooth kernels are bounded in the homogeneous Hardy–Sobolev spaces H1,p(Rd) when p > d/(d + 1). This range of exponents is sharp. As a byproduct of the ...

Vectorvalued operators, optimal weighted estimates and the $C_p$ condition
(201809)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 ...

Proof of an extension of E. Sawyer's conjecture about weighted mixed weaktype estimates
(201809)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
(20180817)In this paper we provide weighted estimates for rough operators, including rough homogeneous singular integrals $T_\Omega$ with $\Omega\in L^\infty(\mathbb{S}^{n1})$ and the BochnerRiesz multiplier at the critical index ...

Borderline Weighted Estimates for Commutators of Singular Integrals
(20160701)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}\f ...

A note on the offdiagonal MuckenhouptWheeden conjecture
(20160701)We obtain the offdiagonal MuckenhouptWheeden conjecture for CalderónZygmund operators. Namely, given $1 < p < q < \infty$ and a pair of weights $(u; v)$, if the HardyLittlewood maximal function satisfies the following ...

$A_1$ theory of weights for rough homogeneous singular integrals and commutators
(20160701)Quantitative $A_1A_\infty$ 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 ...

Three Observations on Commutators of Singular Integral Operators with BMO Functions
(20160701)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\, : ...

Reverse Hölder Property for Strong Weights and General Measures
(20160630)We present dimensionfree reverse Hölder inequalities for strong $A^{\ast}_p$ weights, $1 \le p < \infty$. We also provide a proof for the full range of local integrability of $A^{\ast}_1$ weights. The common ingredient ...

Quantitative weighted mixed weaktype inequalities for classical operators
(20160630)We improve on several mixed weak type inequalities both for the HardyLittlewood maximal function and for CalderónZygmund operators. These type of inequalities were considered by Muckenhoupt and Wheeden and later on by ...

Mixed weak type estimates: Examples and counterexamples related to a problem of E. Sawyer
(20160101)In this paper we study mixed weighted weaktype inequal ities for families of functions, which can be applied to study classic operators in harmonic analysis. Our main theorem extends the key result from [CMP2].