The Bohr–Bohnenblust–Hille theorem states that the width of the
strip in the complex plane on which an ordinary Dirichlet series
converges uniformly but not absolutely is less than or equal to
,
and this estimate is optimal. Equivalently, the supremum of the
absolute convergence abscissas of all Dirichlet series in the Hardy space
equals
.
By a surprising fact of Bayart the same result holds true if
is replaced by
any Hardy space
,
,
of Dirichlet series. For Dirichlet series with coefficients in a Banach space
the maximal width of Bohr’s strips depend on the geometry of
; Defant,
García, Maestre and Pérez-García proved that such maximal width equals
, where
denotes the
maximal cotype of
.
Equivalently, the supremum over the absolute convergence
abscissas of all Dirichlet series in the vector-valued Hardy space
equals
.
In this article we show that this result remains true if
is replaced by
the larger class
,
.