The ℓs\ell^s-boundedness of a family of integral operators on UMD Banach function spaces

We prove the $\ell^s$-boundedness of a family of integral operators with an operator-valued kernel on UMD Banach function spaces. This generalizes and simplifies earlier work by Gallarati, Veraar and the author, where the $\ell^s$-boundedness of this family of integral operators was shown on Lebesgue spaces. The proof is based on a characterization of $\ell^s$-boundedness as weighted boundedness by Rubio de Francia.Comment: 13 pages. Generalization of arXiv:1410.665

Factorization of Operators Through Orlicz Spaces

[EN] We study factorization of operators between quasi-Banach spaces. We prove the equivalence between certain vector norm inequalities and the factorization of operators through Orlicz spaces. As a consequence, we obtain the Maurey-Rosenthal factorization of operators into L-p-spaces. We give several applications. In particular, we prove a variant of Maurey's Extension Theorem.The research of the first author was supported by the National Science Centre (NCN), Poland, Grant No. 2011/01/B/ST1/06243. The research of the second author was supported by Ministerio de Economia y Competitividad, Spain, under project #MTM2012-36740-C02-02Mastylo, M.; Sánchez Pérez, EA. (2017). Factorization of Operators Through Orlicz Spaces. Bulletin of the Malaysian Mathematical Sciences Society. 40(4):1653-1675. https://doi.org/10.1007/s40840-015-0158-5S16531675404Calderón, A.P.: Intermediate spaces and interpolation, the complex method. Stud. Math. 24, 113–190 (1964)Davis, W.J., Garling, D.J.H., Tomczak-Jaegermann, N.: The complex convexity of quasi-normed linear spaces. J. Funct. Anal. 55, 110–150 (1984)Defant, A.: Variants of the Maurey–Rosenthal theorem for quasi Köthe function spaces. Positivity 5, 153–175 (2001)Defant, A., Mastyło, M., Michels, C.: Orlicz norm estimates for eigenvalues of matrices. Isr. J. Math. 132, 45–59 (2002)Defant, A., Sánchez Pérez, E.A.: Maurey–Rosenthal factorization of positive operators and convexity. J. Math. Anal. Appl. 297, 771–790 (2004)Defant, A., Sánchez Pérez, E.A.: Domination of operators on function spaces. Math. Proc. Camb. Phil. Soc. 146, 57–66 (2009)Diestel, J.: Sequences and Series in Banach Spaces. Springer, Berlin (1984)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)Dilworth, S.J.: Special Banach lattices and their applications. In: Handbook of the Geometry of Banach Spaces, vol. 1. Elsevier, Amsterdam (2001)Figiel, T., Pisier, G.: Séries alétoires dans les espaces uniformément convexes ou uniformément lisses. Comptes Rendus de l’Académie des Sciences, Paris, Série A 279, 611–614 (1974)Kalton, N.J., Montgomery-Smith, S.J.: Set-functions and factorization. Arch. Math. (Basel) 61(2), 183–200 (1993)Kamińska, A., Mastyło, M.: Abstract duality Sawyer formula and its applications. Monatsh. Math. 151(3), 223–245 (2007)Kantorovich, L.V., Akilov, G.P.: Functional Analysis, 2nd edn. Pergamon Press, Oxford (1982)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)Lozanovskii, G.Ya.: On some Banach lattices IV, Sibirsk. Mat. Z. 14, 140–155 (1973) (in Russian); English transl.: Siberian. Math. J. 14, 97–108 (1973)Lozanovskii, G.Ya.:Transformations of ideal Banach spaces by means of concave functions. In: Qualitative and Approximate Methods for the Investigation of Operator Equations, Yaroslavl, vol. 3, pp. 122–147 (1978) (Russian)Mastyło, M., Szwedek, R.: Interpolative constructions and factorization of operators. J. Math. Anal. Appl. 401, 198–208 (2013)Nikišin, E.M.: Resonance theorems and superlinear operators. Usp. Mat. Nauk 25, 129–191 (1970) (Russian)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal Domain and Integral Extension of Operators acting in Function Spaces. Operator Theory: Adv. Appl., vol. 180. Birkhäuser, Basel (2008)Pisier, G.: Factorization of linear operators and geometry of Banach spaces. CBMS Regional Conference Series in Mathematics, vol. 60. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1986)Reisner, S.: On two theorems of Lozanovskii concerning intermediate Banach lattices, geometric aspects of functional analysis (1986/87). Lecture Notes in Math., vol. 1317, pp. 67–83. Springer, Berlin (1988)Wojtaszczyk, P.: Banach Spaces for Analysts. Cambridge University Press, Cambridge (1991

Health promotion in primary care: How should we intervene? A qualitative study involving both physicians and patients

<p>Abstract</p> <p>Background</p> <p>The effects of tobacco, physical exercise, diet, and alcohol consumption on morbidity and mortality underline the importance of health promotion and prevention (HPP) at the primary health care (PHC) level. Likewise, the deficiencies when putting such policies into practice and assessing their effectiveness are also widely recognised. The objectives of this research were: a) to gain an in-depth understanding of general practitioners' (GPs) and patients' perceptions about HPP in PHC, and b) to define the areas that could be improved in future interventions.</p> <p>Methods</p> <p>Qualitative methodology focussed on the field of health services research. Information was generated on the basis of two GP-based and two patient-based discussion groups, all of which had previously participated in two interventions concerning healthy lifestyle promotion (tobacco and physical exercise). Transcripts and field notes were analysed on the basis of a sociological discourse-analysis model. The results were validated by triangulation between researchers.</p> <p>Results</p> <p>GPs and patients' discourses about HPP in PHC were different in priorities and contents. An overall explanatory framework was designed to gain a better understanding of the meaning of GP-patient interactions related to HPP, and to show the main trends that emerged from their discourses. GPs linked their perceptions of HPP to their working conditions and experience in health services. The dimensions in this case involved the orientation of interventions, the goal of actions, and the evaluation of results. For patients, habits were mainly related to ways of life particularly influenced by close contexts. Health conceptions, their role as individuals, and the orientation of their demands were the most important dimensions in patients' sphere.</p> <p>Conclusions</p> <p>HPP activities in PHC need to be understood and assessed in the context of their interaction with the conditioning trends in health services and patients' social micro-contexts. On the basis of the explanatory framework, three development lines are proposed: the incorporation of new methodological approaches according to the complexity of HPP in PHC; the openness of habit change policies beyond the medical services; and the effective commitments in the medium to long term by the health services themselves at the policy management level.</p

Product factorability of integral bilinear operators on Banach function spaces

[EN] This paper deals with bilinear operators acting in pairs of Banach function spaces that factor through the pointwise product. We find similar situations in different contexts of the functional analysis, including abstract vector lattices¿orthosymmetric maps, C¿-algebras¿zero product preserving operators, and classical and harmonic analysis¿integral bilinear operators. Bringing together the ideas of these areas, we show new factorization theorems and characterizations by means of norm inequalities. The objective of the paper is to apply these tools to provide new descriptions of some classes of bilinear integral operators, and to obtain integral representations for abstract classes of bilinear maps satisfying certain domination properties.The first author was supported by TUBITAK-The Scientific and Technological Research Council of Turkey, Grant No. 2211/E. The second author was supported by Ministerio de Economia y Competitividad (Spain) and FEDER, Grant MTM2016-77054-C2-1-P.Erdogan, E.; Sánchez Pérez, EA.; Gok, O. (2019). Product factorability of integral bilinear operators on Banach function spaces. Positivity. 23(3):671-696. https://doi.org/10.1007/s11117-018-0632-zS671696233Abramovich, Y.A., Kitover, A.K.: Inverses of Disjointness Preserving Operators. American Mathematical Society, Providence (2000)Abramovich, Y.A., Wickstead, A.W.: When each continuous operator is regular II. Indag. Math. (N.S.) 8(3), 281–294 (1997)Alaminos, J., Brešar, M., Extremera, J., Villena, A.R.: Maps preserving zero products. Studia Math. 193(2), 131–159 (2009)Alaminos, J., Brešar, M., Extremera, J., Villena, A.R.: On bilinear maps determined by rank one idempotents. Linear Algebra Appl. 432, 738–743 (2010)Alaminos, J., Extremera, J., Villena, A.R.: Orthogonality preserving linear maps on group algebras. Math. Proc. Camb. Philos. Soc. 158, 493–504 (2015)Ben Amor, F.: On orthosymmetric bilinear maps. Positivity 14, 123–134 (2010)Astashkin, S.V., Maligranda, L.: Structure of Cesàro function spaces: a survey. Banach Center Publ. 102, 13–40 (2014)Beckenstein, E., Narici, L.: A non-Archimedean Stone–Banach theorem. Proc. Am. Math. Soc. 100(2), 242–246 (1987)Bu, Q., Buskes, G., Kusraev, A.G.: Bilinear maps on products of vector lattices: a survey. In: Boulabiar, K., Buskes, G., Triki, A. (eds.) Positivity: Trends in Mathematics, pp. 97–126. Springer, Birkhuser (2007)Buskes, G., van Rooij, A.: Almost f-algebras: commutativity and Cauchy–Schwarz inequality. Positivity 4, 227–231 (2000)Buskes, G., van Rooij, A.: Squares of Riesz spaces. Rocky Mt. J. Math. 31(1), 45–56 (2001)Calabuig, J.M., Delgado, O., Sánchez Pérez, E.A.: Generalized perfect spaces. Indag. Math. (N.S.) 19(3), 359–378 (2008)Calderón, A.P.: Intermediate spaces and interpolation, the complex method. Studia Math. 24, 113–190 (1964)Defant, A.: Variants of the Maurey-Rosenthal theorem for quasi Köthe function spaces. Positivity 5, 153–175 (2001)Delgado Garrido, O., Sánchez Pérez, E.A.: Strong factorizations between couples of operators on Banach function spaces. J. Convex Anal. 20(3), 599–616 (2013)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators, vol. 43. Cambridge University Press, Cambridge (1995)Erdoğan, E., Calabuig, J.M., Sánchez Pérez, E.A.: Convolution-continuous bilinear operators acting in Hilbert spaces of integrable functions. Ann. Funct. Anal. 9(2), 166–179 (2018)Diestel, J., Uhl, J.J.: Vector Measures. American Mathematical Society, Providence (1977)Fremlin, D.H.: Tensor products of Archimedean vector lattices. Am. J. Math. 94, 778–798 (1972)Gillespie, T.A.: Factorization in Banach function spaces. Nederl. Akad. Wetensch. Indag. Math. 43(3), 287–300 (1981)Grafakos, L., Li, X.: Uniform bounds for the bilinear Hilbert transforms I. Ann. Math. 159, 889–933 (2004)Kantorovich, K.L., Akilov, G.P.: Functional Analysis, Nauka, Moscow 1977 (Russian). English transl. Pergamon Press, Oxford, Elmsford, New York (1982)Kolwicz, P., Leśnik, K., Maligranda, L.: Pointwise products of some Banach function spaces and factorization. J. Funct. Anal. 266(2), 616–659 (2014)Kolwicz, P., Leśnik, K.: Topological and geometrical structure of Calderón–Lozanovskii construction. Math. Inequal. Appl. 13(1), 175–196 (2010)Kühn, B.: Banachverbände mit ordnungsstetiger dualnorm. Math. Z. 167(3), 271–277 (1979)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II: Function Spaces, vol. 97. Springer, Berlin (1979)Lozanovskii, G.Ya.: On some Banach lattices. Sibirsk. Mat. Zh. 10, 584-599 (1969)(Russian)English transl. in Siberian Math. J. 10(3), 419-431 (1969)Maligranda, L., Persson, L.E.: Generalized duality of some Banach function spaces. Nederl. Akad. Wetensch. Indag. 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Noise-Driven Phenotypic Heterogeneity with Finite Correlation Time in Clonal Populations

There has been increasing awareness in the wider biological community of the role of clonal phenotypic heterogeneity in playing key roles in phenomena such as cellular bet-hedging and decision making, as in the case of the phage-λ lysis/lysogeny and B. Subtilis competence/vegetative pathways. Here, we report on the effect of stochasticity in growth rate, cellular memory/intermittency, and its relation to phenotypic heterogeneity. We first present a linear stochastic differential model with finite auto-correlation time, where a randomly fluctuating growth rate with a negative average is shown to result in exponential growth for sufficiently large fluctuations in growth rate. We then present a non-linear stochastic self-regulation model where the loss of coherent self-regulation and an increase in noise can induce a shift from bounded to unbounded growth. An important consequence of these models is that while the average change in phenotype may not differ for various parameter sets, the variance of the resulting distributions may considerably change. This demonstrates the necessity of understanding the influence of variance and heterogeneity within seemingly identical clonal populations, while providing a mechanism for varying functional consequences of such heterogeneity. Our results highlight the importance of a paradigm shift from a deterministic to a probabilistic view of clonality in understanding selection as an optimization problem on noise-driven processes, resulting in a wide range of biological implications, from robustness to environmental stress to the development of drug resistance

A promising method for identifying cross-cultural differences in patient perspective: the use of Internet-based focus groups for content validation of new Patient Reported Outcome assessments

OBJECTIVES: This proof of concept (POC) study was designed to evaluate the use of an Internet-based bulletin board technology to aid parallel cross-cultural development of thematic content for a new set of patient-reported outcome measures (PROs). METHODS: The POC study, conducted in Germany and the United States, utilized Internet Focus Groups (IFGs) to assure the validity of new PRO items across the two cultures – all items were designed to assess the impact of excess facial oil on individuals' lives. The on-line IFG activities were modeled after traditional face-to-face focus groups and organized by a common 'Topic' Guide designed with input from thought leaders in dermatology and health outcomes research. The two sets of IFGs were professionally moderated in the native language of each country. IFG moderators coded the thematic content of transcripts, and a frequency analysis of code endorsement was used to identify areas of content similarity and difference between the two countries. Based on this information, draft PRO items were designed and a majority (80%) of the original participants returned to rate the relative importance of the newly designed questions. FINDINGS: The use of parallel cross-cultural content analysis of IFG transcripts permitted identification of the major content themes in each country as well as exploration of the possible reasons for any observed differences between the countries. Results from coded frequency counts and transcript reviews informed the design and wording of the test questions for the future PRO instrument(s). Subsequent ratings of item importance also deepened our understanding of potential areas of cross-cultural difference, differences that would be explored over the course of future validation studies involving these PROs. CONCLUSION: The use of IFGs for cross-cultural content development received positive reviews from participants and was found to be both cost and time effective. The novel thematic coding methodology provided an empirical platform on which to develop culturally sensitive questionnaire content using the natural language of participants. Overall, the IFG responses and thematic analyses provided a thorough evaluation of similarities and differences in cross-cultural themes, which in turn acted as a sound base for the development of new PRO questionnaires