Bohr radii of vector valued holomorphic functions

Multiple summability properties for Cohen ‐summing operators

“…We recall an inequality of Blei, see [, Lemma 2, p. 430], [, Lemma 5.1], [, Lemma 3.1], [, Corollary 2.2], [, Lemma 2.3]. Lemma Let be two finite sets, .…”

Estimates for vector valued Dirichlet polynomials

“…The crucial point there is the so called hypercontractivity of the polynomial Bohnenblust-Hille inequality. We will use a vector valued variant of this inequality [13,Theorem 5.3]. Note that, although the setting here is quite general, when v = id C then q = 2 and r = 1, we recover the scalar result (1.4).…”

Hausdorff–Young-type inequalities for vector-valued Dirichlet series

“…Analogously, Theorem 3.2 shows the corresponding results for type and its hypercontractive homogeneous version. In [10] variants of vector-valued Bonhenblust-Hille inequality with operators are shown to hold for Banach lattices nontrivial cotype. In [5], results regarding monomial convergence sets and multipliers for Hardy spaces were presented for Banach spaces with nontrivial cotype and local unconditional structure or with Fourier cotype.…”

“…Following exactly the same arguments as in[10, Theorem 5.3] (see also[9,.29]) it can be shown that if Y is a cotype q space and v : X → Y is an (r, 1)-summing operator, then there exists a constant C > 0 so thatα v(c α ) qrm q+(m−1)r Y q+(m−1)r qrm ≤ C m sup z∈D n P(z) Xfor every X-valued polynomial P(z) = α c α z α of n variables of degree m. With this at hand, the estimates in [10, Theorem 1.6-(2) and Theorem 5.4-(2)] hold for Banach spaces with cotype q.…”