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.Comment: To appear in the Electronic Research Announcements in Mathematical Science

Harmonic measure and quantitative connectivity: geometric characterization of the Lp-solvability of the Dirichlet problem

It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-$A_\infty$ property) of harmonic measure with respect to surface measure, on the boundary of an open set $\Omega\subset \mathbb{R}^{n+1}$ with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in $\Omega$, with data in $L^p(\partial\Omega)$ for some $p<\infty$. In this paper, we give a geometric characterization of the weak-$A_\infty$ property, of harmonic measure, and hence of solvability of the $L^p$ Dirichlet problem for some finite $p$. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors-David regularity of the boundary) that are natural, and in a certain sense optimal: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors-David bounds); moreover, the examples show that the upper and lower Ahlfors-David bounds are each quantitatively sharp.Comment: This paper is a combination of arXiv:1712.03696 and arXiv:1803.07975 To appear in Invent. Mat

Jardins per a la salut

Facultat de Farmàcia, Universitat de Barcelona. Ensenyament: Grau de Farmàcia. Assignatura: Botànica farmacèutica. Curs: 2014-2015. Coordinadors: Joan Simon, Cèsar Blanché i Maria Bosch.Els materials que aquí es presenten són el recull de les fitxes botàniques de 128 espècies presents en el Jardí Ferran Soldevila de l’Edifici Històric de la UB. Els treballs han estat realitzats manera individual per part dels estudiants dels grups M-3 i T-1 de l’assignatura Botànica Farmacèutica durant els mesos de febrer a maig del curs 2014-15 com a resultat final del Projecte d’Innovació Docent «Jardins per a la salut: aprenentatge servei a Botànica farmacèutica» (codi 2014PID-UB/054). Tots els treballs s’han dut a terme a través de la plataforma de GoogleDocs i han estat tutoritzats pels professors de l’assignatura. L’objectiu principal de l’activitat ha estat fomentar l’aprenentatge autònom i col·laboratiu en Botànica farmacèutica. També s’ha pretès motivar els estudiants a través del retorn de part del seu esforç a la societat a través d’una experiència d’Aprenentatge-Servei, deixant disponible finalment el treball dels estudiants per a poder ser consultable a través d’una Web pública amb la possibilitat de poder-ho fer in-situ en el propi jardí mitjançant codis QR amb un smartphone

Albiglutide and cardiovascular outcomes in patients with type 2 diabetes and cardiovascular disease (Harmony Outcomes): a double-blind, randomised placebo-controlled trial

Background: Glucagon-like peptide 1 receptor agonists differ in chemical structure, duration of action, and in their effects on clinical outcomes. The cardiovascular effects of once-weekly albiglutide in type 2 diabetes are unknown. We aimed to determine the safety and efficacy of albiglutide in preventing cardiovascular death, myocardial infarction, or stroke. Methods: We did a double-blind, randomised, placebo-controlled trial in 610 sites across 28 countries. We randomly assigned patients aged 40 years and older with type 2 diabetes and cardiovascular disease (at a 1:1 ratio) to groups that either received a subcutaneous injection of albiglutide (30–50 mg, based on glycaemic response and tolerability) or of a matched volume of placebo once a week, in addition to their standard care. Investigators used an interactive voice or web response system to obtain treatment assignment, and patients and all study investigators were masked to their treatment allocation. We hypothesised that albiglutide would be non-inferior to placebo for the primary outcome of the first occurrence of cardiovascular death, myocardial infarction, or stroke, which was assessed in the intention-to-treat population. If non-inferiority was confirmed by an upper limit of the 95% CI for a hazard ratio of less than 1·30, closed testing for superiority was prespecified. This study is registered with ClinicalTrials.gov, number NCT02465515. Findings: Patients were screened between July 1, 2015, and Nov 24, 2016. 10 793 patients were screened and 9463 participants were enrolled and randomly assigned to groups: 4731 patients were assigned to receive albiglutide and 4732 patients to receive placebo. On Nov 8, 2017, it was determined that 611 primary endpoints and a median follow-up of at least 1·5 years had accrued, and participants returned for a final visit and discontinuation from study treatment; the last patient visit was on March 12, 2018. These 9463 patients, the intention-to-treat population, were evaluated for a median duration of 1·6 years and were assessed for the primary outcome. The primary composite outcome occurred in 338 (7%) of 4731 patients at an incidence rate of 4·6 events per 100 person-years in the albiglutide group and in 428 (9%) of 4732 patients at an incidence rate of 5·9 events per 100 person-years in the placebo group (hazard ratio 0·78, 95% CI 0·68–0·90), which indicated that albiglutide was superior to placebo (p&lt;0·0001 for non-inferiority; p=0·0006 for superiority). The incidence of acute pancreatitis (ten patients in the albiglutide group and seven patients in the placebo group), pancreatic cancer (six patients in the albiglutide group and five patients in the placebo group), medullary thyroid carcinoma (zero patients in both groups), and other serious adverse events did not differ between the two groups. There were three (&lt;1%) deaths in the placebo group that were assessed by investigators, who were masked to study drug assignment, to be treatment-related and two (&lt;1%) deaths in the albiglutide group. Interpretation: In patients with type 2 diabetes and cardiovascular disease, albiglutide was superior to placebo with respect to major adverse cardiovascular events. Evidence-based glucagon-like peptide 1 receptor agonists should therefore be considered as part of a comprehensive strategy to reduce the risk of cardiovascular events in patients with type 2 diabetes. Funding: GlaxoSmithKline

Weights, Extrapolation and the Theory of Rubio de Francia

This book provides a systematic development of the Rubio de Francia theory of extrapolation, its many generalizations and its applications to one and two-weight norm inequalities. The book is based upon a new and elementary proof of the classical extrapolation theorem that fully develops the power of the Rubio de Francia iteration algorithm. This technique allows us to give a unified presentation of the theory and to give important generalizations to Banach function spaces and to two-weight inequalities. We provide many applications to the classical operators of harmonic analysis to illustrat

The weak-A∞ property of harmonic and p-harmonic measures implies uniform rectifiability

Let E\subset \ree, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set\Omega:= \ree\setminus E, implies uniform rectifiability of $E$. More generally, we establish a similar result for the Riesz measure, $p$-harmonic measure,associated to the $p$-Laplace operator, 1<p<\infty

On the A∞ condition for elliptic operators in 1-sided nontangentially accessible domains satisfying the capacity density condition

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