Bohr’s radii and strips – a microscopic and a macroscopic view

The Bohr-Bohnenblust-Hille theorem states that the largest possible width $S$ of the strip in the complex plane on which a Dirichlet series $\sum_n a_n 1/n^s$ converges uniformly but not absolutely, equals $1/2$. In fact Bohr in 1913 proved that $S \leq 1/2$, and asked for equality. The general theory of Dirichlet series during this time was one of the most fashionable topics in analysis, and Bohr's so-called \textit{absolute convergence problem} was very much in the focus. In this context Bohr himself discovered several deep connections of Dirichlet series and power series (holomorphic functions) in infinitely many variables, and as a sort of by-product he found his famous power series theorem. Finally, Bohnenblust and Hille in 1931 in a rather ingenious fashion answered the absolute convergence problem in the positive. In recent years many authors revisited the work of Bohr, Bohnenblust and Hille -- improving this work but also extending it to more general settings, for example to Dirichlet series with coefficients in Banach spaces. The aim of this article is to report on parts of this new development

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

The Gonadotropin-Releasing Hormone (GnRH) Neuronal Population Is Normal in Size and Distribution in GnRH-Deficient and GnRH Receptor-Mutant Hypogonadal Mice

Hypothalamic GnRH neurons are essential for initiation and regulation of reproductive function. In addition to pituitary gonadotrope stimulation, activity of GnRH through its receptor (GnRHR) has been suggested to include autocrine regulation of the GnRH neuron. Two hypogonadal mouse strains, the Gnrh1 mutant (hpg) mice and Gnrhr mutant mice were used to investigate the potential role of GnRH signaling in the proper development and maintenance of GnRH neurons. Immunocytochemical analysis of heterozygous hpg mice revealed a GnRH neuron population that was normal in size and distribution, indicating no effect from reduced Gnrh1 gene dosage on the neurons themselves. To visualize GnRH neurons in homozygous GnRH-deficient hpg mice, heterozygous hpg mice were crossed with GnRH-green fluorescent protein (GFP) transgenic mice with targeted expression of the GFP reporter gene in GnRH neurons. Analysis of forebrains of homozygous hpg/GFP-positive mice immunostained for GFP revealed a normal population size and appropriate distribution of GnRH neurons in hpg mice, with immunoreactive neuronal processes present at the median eminence. Similarly, adult mice deficient in functional GnRHR possessed a full complement of GnRH neurons in the basal forebrain that was indistinguishable from the distribution of GnRH neurons in their wild-type counterparts. Moreover, hpg/GFP neurons retained the ability to generate spontaneous bursts of action potential firing activity, suggesting that GnRH peptide is not required for this function. These data establish that autocrine-paracrine GnRH-signaling is not a prerequisite for the developmental migration of GnRH neurons into the brain or for the projection of GnRH neurosecretory axons