Convergence of Dirichlet polynomials in Banach spaces

Abstract

Abstract. Recent results on Dirichlet seriesn a n 1 n s , s ∈ C, with coefficients a n in an infinite dimensional Banach space X show that the maximal width of uniform but not absolute convergence coincides for Dirichlet series and for m-homogeneous Dirichlet polynomials. But a classical non-trivial fact due to Bohnenblust and Hille shows that if X is one dimensional, this maximal width heavily depends on the degree m of the Dirichlet polynomials. We carefully analyze this phenomenon, in particular in the setting of p -spaces.

A new multilinear insight on Littlewood's 4/3-inequality

“…In the same way as (2) and (3) play an important role in [4,19], so also using Theorem 1 we are able to give a complete answer in the p -case in [15]: For 1 p q ∞ we consider mhomogeneous Dirichlet polynomials a n /n s in q whose coefficients a n ∈ p and for each one of them the difference between the abscissa of uniform convergence in p and that of absolute convergence in q . We then define T m (p, q) to be the maximal width of these strips in C. This number somehow measures how much the summability of a homogeneous Dirichlet polynomial in p improves when we move from p to a bigger q .…”

“…Although we believe that our main results from Theorems 1 and 4 are of independent interest we here want to sketch the application which originally motivated these results, and which will be presented in the forthcoming paper [15]. Bohr showed in [5] that the width of the strip in C on which a Dirichlet series a n /n s , s ∈ C, converges uniformly but not absolutely, is at most 1/2, and Bohnenblust and Hille in [4] were able to prove that this bound is even optimal.…”

Coordinatewise multiple summing operators in Banach spaces

“…Both inequalities, the Bohnenblust-Hille inequality and Littlewood's 4/3-inequality, had and have deep applications in various fields of analysis, as for example in operator theory in Banach spaces, Fourier and harmonic analysis, complex analysis in finitely and infinitely many variables, and analytic number theory (see e.g. the monographs [3,20,28,29], or the more recent articles [2,6,7,9,12,13,15,16,19,21,31]). …”

Estimates for vector valued Dirichlet polynomials

“…The study of the width of Bohr's strips for such objects was initiated in [11] and continued in [15]. Given a operator v : X → Y between two Banach spaces we define the number…”

“…In [15] this phenomenon was analysed for operators on the p -spaces. We have in [12,Corollary 5.7] and [15,Theorem 1.1] …”