We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if $f\left(s\right)={\sum }_{n=1}^{\infty }aₙn^{-s}$ with ${\left|\right|f\left|\right|}_{\infty }:=su{p}_{\Re s>0}\left|f\left(s\right)\right|<\infty$, then ${\sum }_{n=1}^{\infty }\left|aₙ\right|n^{-2}{\le \left|\right|f\left|\right|}_{\infty }$ and even slightly better, and ${\sum }_{n=1}^{\infty }\left|aₙ\right|n^{-1/2}{\le C\left|\right|f\left|\right|}_{\infty }$, C being an absolute constant.

We consider sets in the real line that have Littlewood-Paley properties LP(p) or LP and study the following question: How thick can these sets be?

We study convolution operators bounded on the non-normable Lorentz spaces $L^{1,q}$ of the real line and the torus. Here 0 < q < 1. On the real line, such an operator is given by convolution with a discrete measure, but on the torus a convolutor can also be an integrable function. We then give some necessary and some sufficient conditions for a measure or a function to be a convolutor on $L^{1,q}$. In particular, when the positions of the atoms of a discrete measure are linearly independent over the rationals,...

By a Fourier multiplier technique on Cantor-like Abelian groups with characters of finite order, the norms from L² into $L^{2l}$ of certain embeddings of character sums are computed. It turns out that the orders of the characters are immaterial as soon as they are at least four.