Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators

This is the third part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. For $L$ in some class of elliptic operators, we study weighted norm $L^p$ inequalities for singular 'non-integral' operators arising from $L$ ; those are the operators $\phi(L)$ for bounded holomorphic functions $\phi$, the Riesz transforms $\nabla L^{-1/2}$ (or $(-\Delta)^{1/2}L^{-1/2}$) and its inverse $L^{1/2}(-\Delta)^{-1/2}$, some quadratic functionals $g\_{L}$ and $G\_{L}$ of Littlewood-Paley-Stein type and also some vector-valued inequalities such as the ones involved for maximal $L^p$-regularity. For each, we obtain sharp or nearly sharp ranges of $p$ using the general theory for boundedness of Part I and the off-diagonal estimates of Part II. We also obtain commutator results with BMO functions.Comment: 38 pages. Third of 4 paper

Weighted norm inequalities for fractional operators]

We prove weighted norm inequalities for fractional powers of elliptic operators together with their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The method relies upon a good-$\lambda$ method that does not use any size or smoothness estimates for the kernels.Comment: accepted in Indiana University Mathematicla Journal. A thorough reorganisation has been done on suggestions from the refere

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part II: Off-diagonal estimates on spaces of homogeneous type

This is the second part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. We consider a substitute to the notion of pointwise bounds for kernels of operators which usually is a measure of decay. This substitute is that of off-diagonal estimates expressed in terms of local and scale invariant $L^p-L^q$ estimates. We propose a definition in spaces of homogeneous type that is stable under composition. It is particularly well suited to semigroups. We study the case of semigroups generated by elliptic operators.Comment: 40 pages. Second of 4 papers. Can be read independentl

Vertical versus conical square functions

We study the difference between vertical and conical square functions in the abstract and also in the specific case where the square functions come from an elliptic operator.Comment: 21 page

Weighted norm inequalities, off-diagonal estimates and elliptic operators

We give an overview of the generalized Calder\'on-Zygmund theory for "non-integral" singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted estimates for such operators and their commutators with \BMO functions. $L^p-L^q$ off-diagonal estimates when $p\le q$ play an important role and we present them. They are particularly well suited to the semigroups generated by second order elliptic operators and the range of exponents $(p,q)$ rules the $L^p$ theory for many operators constructed from the semigroup and its gradient. Such applications are summarized.Comment: survey for the El Escorial 2008 proceeding

Lp self improvement of generalized Poincaré inequalities on spaces of homogeneous type

International audienceIn this paper we study self-improving properties in the scale of Lebesgue spaces of generalized Poincaré inequalities in spaces of homogeneous type. In contrast with the classical situation, the oscillations involve approximation of the identities or semigroups whose kernels decay fast enough and the resulting estimates take into account their lack of localization. The techniques used do not involve any classical Poincaré or Sobolev-Poincaré inequalities and therefore they can be used in general settings where these estimates do not hold or are unknown. We apply our results to the case of Riemannian manifolds with doubling volume form and assuming Gaussian upper bounds for the heat kernel of the semigroup $e^{-t\,\Delta}$ with $\Delta$ being the Laplace-Beltrami operator. We obtain generalized Poincaré inequalities with oscillations that involve the semigroup $e^{-t\,\Delta}$ and with right hand sides containing either $\nabla$ or $\Delta^{1/2}$

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<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 (<1%) deaths in the placebo group that were assessed by investigators, who were masked to study drug assignment, to be treatment-related and two (<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