In this paper, we prove some of Rubio de Francia's extrapolation results for the class B p of wei... more In this paper, we prove some of Rubio de Francia's extrapolation results for the class B p of weights for which the Hardy operator is bounded on L p (w) restricted to decreasing functions. Applications to the boundedness of operators on L p dec (w) are given. We also present an extension to the B ∞ case and some connections with classical A p theory.
Journal of Fourier Analysis and Applications, 2021
We present new estimates in the setting of weighted Lorentz spaces for important operators in Har... more We present new estimates in the setting of weighted Lorentz spaces for important operators in Harmonic Analysis such as sparse operators, Bochner-Riesz at the critical index, Hörmander multipliers and rough singular integrals among others. Keywords Weights • Restricted weak type Rubio de Francia extrapolation • Weighted Lorentz spaces • Hardy-Littlewood maximal operator Mathematics Subject Classification 42B99 • 46E30 1 p < ∞ .
Memoirs of the American Mathematical Society, 2007
, José Antonio Raposo, the second named author, died when he was only 39. He had just received hi... more , José Antonio Raposo, the second named author, died when he was only 39. He had just received his Ph.D. degree a few months earlier, under our supervision. He was really happy for all the new projects he had for the future, and so were we, since he was an extraordinary mathematician, and a very valuable friend. This work is an updated version of his thesis, written as a self-contained text, with most of the motivations, examples and applications available in the literature. We want to thank all the people who have encouraged us to write this book, and specially José Antonio's family. We also thank Joan Cerdà who has read the whole manuscript and has given us many good advises, improving the final version of these notes.
Boundedness of integral operators on decreasing functions
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2015
We continue the study of the boundedness of the operatoron the set of decreasing functions inLp(w... more We continue the study of the boundedness of the operatoron the set of decreasing functions inLp(w). This operator was first introduced by Braverman and Lai and also studied by Andersen, and although the weighted weak-type estimatewas completely solved, the characterization of the weightswsuch thatis bounded is still open for the case in whichp> 1. The solution of this problem will have applications in the study of the boundedness on weighted Lorentz spaces of important operators in harmonic analysis.
In this paper, we study how the limited and weakly compact properties of operators are preserved ... more In this paper, we study how the limited and weakly compact properties of operators are preserved by interpolation of the real method for infinite families of Banach spaces introduced by Carro in Studia Math. 109 (1994). We apply these results to the case of Sparr, Fernández and Cobos-Peetre methods of interpolation for finite families.
If $P,Q:[0,\infty)\to$ are increasing functions and $T$ is the Calderón operator defined on posit... more If $P,Q:[0,\infty)\to$ are increasing functions and $T$ is the Calderón operator defined on positive or decreasing functions, then optimal modular inequalities $\int P(Tf)\leq C\int Q(f)$ are proved. If $P=Q$, the condition on $P$ is both necessary and sufficient for the modular inequality. In addition, we establish general interpolation theorems for modular spaces.
We prove some extrapolation results for operators bounded on radial L p functions with p ∈ (p0, p... more We prove some extrapolation results for operators bounded on radial L p functions with p ∈ (p0, p1) and deduce some endpoint estimates. We apply our results to prove the almost everywhere convergence of the spherical partial Fourier integrals and to obtain estimates on maximal Bochner-Riesz type operators acting on radial functions in several weighted spaces.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2002
The work of Coifman and Weiss concerning Hardy spaces on spaces of homogeneous type gives, as a p... more The work of Coifman and Weiss concerning Hardy spaces on spaces of homogeneous type gives, as a particular case, a definition of Hp(ZN) in terms of an atomic decomposition.Other characterizations of these spaces have been studied by other authors, but it was an open question to see if they can be defined, as it happens in the classical case, in terms of a maximal function or via the discrete Riesz transforms.In this paper, we give a positive answer to this question.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1999
This work connects the theory of commutators with analytic families of operators in abstract inte... more This work connects the theory of commutators with analytic families of operators in abstract interpolation theory. Our main result asserts that if {Lξ}0≤Reξ≤1 is an analytic family of operators satisfying some conditions, then [Lθ,Ω] +(Lξ)′(θ): Āθ→ Bθ is bounded. From this, we can deduce the boundedness of the commutator .
In this paper, we prove some of Rubio de Francia's extrapolation results for the class B p of wei... more In this paper, we prove some of Rubio de Francia's extrapolation results for the class B p of weights for which the Hardy operator is bounded on L p (w) restricted to decreasing functions. Applications to the boundedness of operators on L p dec (w) are given. We also present an extension to the B ∞ case and some connections with classical A p theory.
Journal of Fourier Analysis and Applications, 2021
We present new estimates in the setting of weighted Lorentz spaces for important operators in Har... more We present new estimates in the setting of weighted Lorentz spaces for important operators in Harmonic Analysis such as sparse operators, Bochner-Riesz at the critical index, Hörmander multipliers and rough singular integrals among others. Keywords Weights • Restricted weak type Rubio de Francia extrapolation • Weighted Lorentz spaces • Hardy-Littlewood maximal operator Mathematics Subject Classification 42B99 • 46E30 1 p < ∞ .
Memoirs of the American Mathematical Society, 2007
, José Antonio Raposo, the second named author, died when he was only 39. He had just received hi... more , José Antonio Raposo, the second named author, died when he was only 39. He had just received his Ph.D. degree a few months earlier, under our supervision. He was really happy for all the new projects he had for the future, and so were we, since he was an extraordinary mathematician, and a very valuable friend. This work is an updated version of his thesis, written as a self-contained text, with most of the motivations, examples and applications available in the literature. We want to thank all the people who have encouraged us to write this book, and specially José Antonio's family. We also thank Joan Cerdà who has read the whole manuscript and has given us many good advises, improving the final version of these notes.
Boundedness of integral operators on decreasing functions
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2015
We continue the study of the boundedness of the operatoron the set of decreasing functions inLp(w... more We continue the study of the boundedness of the operatoron the set of decreasing functions inLp(w). This operator was first introduced by Braverman and Lai and also studied by Andersen, and although the weighted weak-type estimatewas completely solved, the characterization of the weightswsuch thatis bounded is still open for the case in whichp> 1. The solution of this problem will have applications in the study of the boundedness on weighted Lorentz spaces of important operators in harmonic analysis.
In this paper, we study how the limited and weakly compact properties of operators are preserved ... more In this paper, we study how the limited and weakly compact properties of operators are preserved by interpolation of the real method for infinite families of Banach spaces introduced by Carro in Studia Math. 109 (1994). We apply these results to the case of Sparr, Fernández and Cobos-Peetre methods of interpolation for finite families.
If $P,Q:[0,\infty)\to$ are increasing functions and $T$ is the Calderón operator defined on posit... more If $P,Q:[0,\infty)\to$ are increasing functions and $T$ is the Calderón operator defined on positive or decreasing functions, then optimal modular inequalities $\int P(Tf)\leq C\int Q(f)$ are proved. If $P=Q$, the condition on $P$ is both necessary and sufficient for the modular inequality. In addition, we establish general interpolation theorems for modular spaces.
We prove some extrapolation results for operators bounded on radial L p functions with p ∈ (p0, p... more We prove some extrapolation results for operators bounded on radial L p functions with p ∈ (p0, p1) and deduce some endpoint estimates. We apply our results to prove the almost everywhere convergence of the spherical partial Fourier integrals and to obtain estimates on maximal Bochner-Riesz type operators acting on radial functions in several weighted spaces.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2002
The work of Coifman and Weiss concerning Hardy spaces on spaces of homogeneous type gives, as a p... more The work of Coifman and Weiss concerning Hardy spaces on spaces of homogeneous type gives, as a particular case, a definition of Hp(ZN) in terms of an atomic decomposition.Other characterizations of these spaces have been studied by other authors, but it was an open question to see if they can be defined, as it happens in the classical case, in terms of a maximal function or via the discrete Riesz transforms.In this paper, we give a positive answer to this question.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1999
This work connects the theory of commutators with analytic families of operators in abstract inte... more This work connects the theory of commutators with analytic families of operators in abstract interpolation theory. Our main result asserts that if {Lξ}0≤Reξ≤1 is an analytic family of operators satisfying some conditions, then [Lθ,Ω] +(Lξ)′(θ): Āθ→ Bθ is bounded. From this, we can deduce the boundedness of the commutator .
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Papers by María J. Carro