The p-Bohr radius of a Banach space

• Autores: Óscar Blasco de la Cruz
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 68, Fasc. 1, 2017, págs. 87-100
• Idioma: inglés
• DOI: 10.1007/s13348-016-0181-3
• Texto completo no disponible (Saber más ...)
• Resumen

  • Following the scalar-valued case considered by Djakow and Ramanujan (A remark on Bohr’s theorem and its generalizations 14:175–178, 2000) we introduce, for each complex Banach space X and each \[1 \leq p < \infty\], the p-Bohr radius of X as the value \[r_p(X) = \sup \left\{ r > 0 : \sum_{n=0}^{\infty} \| x_n \|_r^{np} r^n \leq \sup_{|z|<1} \| f(z) \|^p \right\}\] where x_n\in X for each n\in \mathbb {N}\cup \{0\} and f(z)=\sum _{n=0}^\infty x_nz^n\in H^\infty (\mathbb {D},X). We show that a complex (possibly infinite dimensional) Banach space X is p-uniformly \mathbb {C}-convex for p\ge 2 if and only if \[r_p(X) > 0\]. We study the p-Bohr radius of the Lebesgue spaces L^q(\mu ) for different values of p and q. In particular we show that r_p(L^q(\mu ))=0 whenever \[p < 2\] and dim(L^q(\mu ))\ge 2 and r_p(L^q(\mu ))=1 whenever p\ge 2 and p'\le q\le p. We also provide some lower estimates for r_2(L^q(\mu )) for the values \[1 \leq q < 2\].