Estimates for vector valued Dirichlet polynomials

[EN] We estimate the -norm of finite Dirichlet polynomials with coefficients in a Banach space. Our estimates quantify several recent results on Bohr's strips of uniform but non absolute convergence of Dirichlet series in Banach spaces.A. Defant and P. Sevilla-Peris were supported by MICINN Project MTM2011-22417.Defant, A.; Schwarting, U.; Sevilla Peris, P. (2014). Estimates for vector valued Dirichlet polynomials. Monatshefte f�r Mathematik. 175(1):89-116. https://doi.org/10.1007/s00605-013-0600-4S891161751Balasubramanian, R., Calado, B., Queffélec, H.: The Bohr inequality for ordinary Dirichlet series. Studia Math. 175(3), 285–304 (2006)Bayart, F.: Hardy spaces of Dirichlet series and their composition operators. Monatsh. Math. 136(3), 203–236 (2002)Bennett, G.: Inclusion mappings between $$l^{p}$$ l p spaces. J. Funct. Anal. 13, 20–27 (1973)Bohnenblust, H.F., Hille, E.: On the absolute convergence of Dirichlet series. Ann. Math. (2) 32(3), 600–622 (1931)Bohr, H.: Über die Bedeutung der Potenzreihen unendlich vieler Variablen in der Theorie der Dirichlet–schen Reihen $$\sum \frac{a_n}{n^s}$$ ∑ a n n s . Nachr. Ges. Wiss. Göttingen Math. Phys. Kl., Heft 4, 441–488 (1913)Bohr, H.: Über die gleichmäßige Konvergenz Dirichletscher Reihen. J. Reine Angew. Math. 143, 203–211 (1913)Carl, B.: Absolut- $$(p,\,1)$$ ( p , 1 ) -summierende identische Operatoren von $$l_{u}$$ l u in $$l_{v}$$ l v . Math. Nachr. 63, 353–360 (1974)Carlson, F.: Contributions à la théorie des séries de Dirichlet. Note i. Ark. fö”r Mat., Astron. och Fys. 16(18), 1–19 (1922)de la Bretèche, R.: Sur l’ordre de grandeur des polynômes de Dirichlet. Acta Arith. 134(2), 141–148 (2008)Defant, A., Frerick, L., Ortega-Cerdà, J., Ounaïes, M., Seip, K.: The Bohnenblust–Hille inequality for homogeneous polynomials is hypercontractive. Ann. Math. (2) 174(1), 485–497 (2011)Defant, A., García, D., Maestre, M., Pérez-García, D.: Bohr’s strip for vector valued Dirichlet series. Math. Ann. 342(3), 533–555 (2008)Defant, A., García, D., Maestre, M., Sevilla-Peris, P.: Bohr’s strips for Dirichlet series in Banach spaces. Funct. Approx. Comment. Math. 44(part 2), 165–189 (2011)Defant, A., Maestre, M., Schwarting, U.: Bohr radii of vector valued holomorphic functions. Adv. Math. 231(5), 2837–2857 (2012)Defant, A., Popa, D., Schwarting, U.: Coordinatewise multiple summing operators in Banach spaces. J. Funct. Anal. 259(1), 220–242 (2010)Defant, A., Sevilla-Peris, P.: Convergence of Dirichlet polynomials in Banach spaces. Trans. Am. Math. Soc. 363(2), 681–697 (2011)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge Studies in Advanced Mathematics, vol. 43. Cambridge University Press, Cambridge (1995)Harris, L.A.: Bounds on the derivatives of holomorphic functions of vectors. In: Analyse fonctionnelle et applications (Comptes Rendus Colloq. Analyse, Inst. Mat., Univ. Federal Rio de Janeiro, Rio de Janeiro, 1972), pp. 145–163. Actualités Aci. Indust., No. 1367. Hermann, Paris (1975)Hedenmalm, H., Lindqvist, P., Seip, K.: A Hilbert space of Dirichlet series and systems of dilated functions in $$L^2(0,1)$$ L 2 ( 0 , 1 ) . Duke Math. J. 86(1), 1–37 (1997)Kahane, J.-P.: Some Random Series of Functions. Cambridge Studies in Advanced Mathematics, vol. 5, 2nd edn. Cambridge University Press, Cambridge (1985)Konyagin, S.V., Queffélec, H.: The translation $$\frac{1}{2}$$ 1 2 in the theory of Dirichlet series. Real Anal. Exch. 27(1):155–175 (2001/2002)Kwapień, S.: Some remarks on $$(p,\, q)$$ ( p , q ) -absolutely summing operators in $$l_{p}$$ l p -spaces. Studia Math. 29, 327–337 (1968)Ledoux, M., Talagrand, M.: Probability in Banach Spaces: Isoperimetry and Processes, reprint of the 1991 edn. Classics in Mathematics. Springer, Berlin (2011)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. I. Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92. Springer, Berlin (1977)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. II, Function Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97. Springer, Berlin (1979)Maurizi, B., Queffélec, H.: Some remarks on the algebra of bounded Dirichlet series. J. Fourier Anal. Appl. 16, 676–692 (2010)Prachar, K.: Primzahlverteilung. Springer, Berlin (1957)Queffélec, H.: H. Bohr’s vision of ordinary Dirichlet series; old and new results. J. Anal. 3, 43–60 (1995)Tomczak-Jaegermann, N.: Banach–Mazur Distances and Finite-Dimensional Operator Ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 38. Longman Scientific & Technical, Harlow (1989

Mitochondrial Structure, Function and Dynamics Are Temporally Controlled by c-Myc

Although the c-Myc (Myc) oncoprotein controls mitochondrial biogenesis and multiple enzymes involved in oxidative phosphorylation (OXPHOS), the coordination of these events and the mechanistic underpinnings of their regulation remain largely unexplored. We show here that re-expression of Myc in myc−/− fibroblasts is accompanied by a gradual accumulation of mitochondrial biomass and by increases in membrane polarization and mitochondrial fusion. A correction of OXPHOS deficiency is also seen, although structural abnormalities in electron transport chain complexes (ETC) are not entirely normalized. Conversely, the down-regulation of Myc leads to a gradual decrease in mitochondrial mass and a more rapid loss of fusion and membrane potential. Increases in the levels of proteins specifically involved in mitochondrial fission and fusion support the idea that Myc affects mitochondrial mass by influencing both of these processes, albeit favoring the latter. The ETC defects that persist following Myc restoration may represent metabolic adaptations, as mitochondrial function is re-directed away from producing ATP to providing a source of metabolic precursors demanded by the transformed cell