Two-weight, weak-type norm inequalities for fractional integrals, Calderón-Zygmund operators and commutators

We give Ap-type conditions which are sufficient for the two-weight, weak-type (p, p) inequalities for fractional integral operators, Calderón-Zygmund operators and commutators. For fractional integral operators, this solves a problem posed by Sawyer and Wheeden. At the heart of all of our proofs is an inequality relating the Hardy-Littlewood maximal function and the sharp maximal function which is strongly reminiscent of the good-λ inequality of Fefferman and Stein.Ford FoundationDirección General de Investigación Científica y Técnic

Two weight extrapolation via the maximal operator

We give several extrapolation theorems for pairs of weights of the form (w, Mkw) and (w, (Mw/w)r w), where w is any non-negative function, r>1, and Mk is the kth iterate of the Hardy–Littlewood maximal operator. As an application we show that our results can be used to extend and sharpen results for square functions and singular integral operators by Chang et al. (1985, Comment. Math. Helv.60, 217–246), Chanillo and Wheeden (1987, Indiana Univ. Math. J.36, 277–294), Wilson (1987, Duke Math. J.55, 879–887; 1989, Trans. Amer. Math. Soc.314, 661–692; 1989, Illinois J. Math.33, 361–366), and Uchiyama (1995, Studia Math.115, 135–149). In the process we prove a conjecture due to Wilson.Dirección General de Investigación Científica y Técnic

Two-weight, weak-type norm inequalities for singular integral operators

We give a sufficient condition for singular integral operators and, more generally, Calder´on-Zygmund operators to satisfy the weak (p, p) inequality u({x ∈ R n : |T f(x)| > t}) ≤ C / tp Z Rn |f|p v dx, 1 < p < ∞. Our condition is an Ap-type condition in the scale of Orlicz spaces: kukL(log L) p−1+δ,Q 1 |Q| Z Q v −p 0/p dx p/p0 ≤ K 0. This conditions is stronger than the Ap condition and is sharp since it fails when δ = 0.Dirección General de Investigación Científica y Técnic

Definición de patrones electrocardiográficos para su reconocimiento en una aplicación informática

Aprendizaje de la electrocardiografía para los alumnos de 3º y 4º de Medicina a partir de una biblioteca de electrocardiografía tutelada, creada en un proyecto de innovación y mejora de la calidad docente de 2013. Se trata de detallar los patrones electrocardiográficos reales de todas las alteraciones electrocardiográficas descritas para que puedan ser llevados a una aplicación gráfica de identificación para dispositivos móviles que diagnostique los electrocardiogramas mediante su escaneo

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 $T$.Comment: We improve different parts of the first version, in particular we show the sharpness of our theorem for the vector-valued maximal functio

Extrapolation from A∞ weights and applications

We generalize the Ap extrapolation theorem of Rubio de Francia to A∞ weights in the context of Muckenhoupt bases. Our result has several important features. First, it can be used to prove weak endpoint inequalities starting from strong-type inequalities, something which is impossible using the classical result. Second, it provides an alternative to the technique of good-λ inequalities for proving Lp norm inequalities relating operators. Third, it yields vector-valued inequalities without having to use the theory of Banach space valued operators. We give a number of applications to maximal functions, singular integrals, potential operators, commutators, multilinear Calder´on-Zygmund operators, and multiparameter fractional integrals. In particular, we give new proofs, which completely avoid the good-λ inequalities, of Coifman’s inequality relating singular integrals and the maximal operator, of the Fefferman-Stein inequality relating the maximal operator and the sharp maximal operator, and the Muckenhoupt-Wheeden inequality relating the fractional integral operator and the fractional maximal operator.Ministerio de Ciencia y Tecnologí

Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture

We prove two-weight norm inequalities for Calderón-Zygmund singular integrals that are sharp for the Hilbert transform and for the Riesz transforms. In addition, we give results for the dyadic square function and for commutators of singular integrals. As an application we give new results for the Sarason conjecture on the product of unbounded Toeplitz operators on Hardy spaces.Stewart Faculty Development Fund (Trinity College)Ministerio de Educación y Cienci

Educación, música y matemáticas: un triángulo afinado en armonía

Establecer una relación entre la música y las matemáticas parte de un interés didáctico que busca como objetivo principal poder usar las cualidades más favorables de una de estas materias, en términos de enseñanza – aprendizaje, para ser aplicadas en la otra. Ambas pueden compartir esta simbiosis. La más buscada (no lo negamos) es aquella que permite aprovechar la buena aceptación, atractivo y motivación que ejerce la música sobre las personas, para aplicarla a la necesidad (menos atractiva) de adquirir conocimientos matemáticos. Para muchos niños pudiera ser de gran ayuda esta relación entre las matemáticas y la música, teniendo en cuenta la dificultad que para algunos supone la disciplina matemática. Hoy tienen a su disposición ayudas pedagógicas que eran inimaginables para su generación predecesora. Estamos, por tanto, en un nuevo proceso evolutivo de la enseñanza. Quizá el más significativo y veloz que jamás haya conocido la escuela. Contribuimos aquí, con este pequeño grano de arena, que esperamos sirva, al menos, para la reflexión y sensibilización del lector interesadoGrado en Educación Primari