Variants of the Maurey-Rosenthal theorem for quasi K"othe function spaces

The Maurey-Rosenthal theorem states that each bounded and linear operator T from a quasi normed space E into some L_p(\nu) which satisfies a certain vector-valued inequality even allows a weighted norm inequality. Continuing the work of Garcia Cuerva and Rubio de Francia we give several scalar and vector-valued variants of this fundamental result within the framework of quasi K"othe function spaces over measure spaces

Complex interpolation of spaces of operators on l_1

Within the theory of complex interpolation and theta-Hilbert spaces we extend classical results of Kwapien on absolutely (r,1)-summing operators on l_1 with values in l_p as well as their natural extensions for mixing operators invented by Maurey. Furthermore, we show that for 1<p<2 every operator T on l_1 with values in theta-type 2 spaces, theta=2/p', is Rademacher p-summing. This is another extension of Kwapien's results, and by an extrapolation procedure a natural supplement to a statement of Pisier.Comment: 15 page

A complex interpolation formula for tensor products of vector-valued Banach function spaces

We prove a complex interpolation formula for the injective tensor product of vector-valued Banach function spaces satisfying certain geometric assumptions. This result unifies results of Kouba, and moreover, our approach offers an alternate proof of Kouba's interpolation formula for scalar-valued Banach function spaces.Comment: 12 page

Hardy spaces of vector-valued Dirichlet series

Given a Banach space $X$ and $1 \leq p \leq \infty$, it is well known that the two Hardy spaces $H_p(\mathbb{T},X)$ ($\mathbb{T}$ the torus) and $H_p(\mathbb{D},X)$ ($\mathbb{D}$ the disk) have to be distinguished carefully. This motivates us to define and study two different types of Hardy spaces $\mathcal{H}_p(X)$ and $\mathcal{H}^+_p(X)$ of Dirichlet series $\sum_n a_n n^{-s}$ with coefficients in $X$. We characterize them in terms of summing operators as well as holomorphic functions in infinitely many variables, and prove that they coincide whenever $X$ has the analytic Radon-Nikod\'{y}m Property. Consequences are, among others, a vector-valued version of the Brother's Riesz Theorem in the infinite-dimensional torus, and an answer to the question when $\mathcal{H}_1(X^{\ast})$ is a dual space.Comment: 24 page

Bennett-Carl inequalities for symmetric Banach sequence spaces and unitary ideals

We prove an abstract interpolation theorem which interpolates the (r,2)-summing and (s,2)-mixing norm of a fixed operator in the image and the range space. Combined with interpolation formulas for spaces of operators we obtain as an application the original Bennett-Carl inequalities for identities acting between Minkowski spaces l_u as well as their analogues for Schatten classes S_u. Furthermore, our techniques motivate a study of Bennett-Carl inequalities within a more general setting of symmetric Banach sequence spaces and unitary ideals.Comment: 17 page

The Levy-Steinitz rearrangement theorem for duals of metrizable spaces

Extending the classical Levy-Steinitz rearrangement theorem, which in turn extended Riemann's theorem, Banaszczyk proved in 1990/93 that a metrizable, locally convex space is nuclear if and only if the domain of sums of every convergent series (i.e. the set of all elements in the space which are sums of a convergent rearrangement of the series) is a translate of a closed subspace of a special form. In this paper we present an apparently complete analysis of the domains of convergent series in duals of metrizable spaces or, more generally, in (DF)-spaces in the sense of Grothendieck

Optimal comparison of PP-norms of Dirichlet Polynomials

Let $1 \leq p < q < \infty$. We show that $$\sup{\frac{\left\| D\right\|_{\mathcal{H}_{q}}}{\left\| D\right\|_{\mathcal{H}_{p}}}} = \exp{\left( \frac{\log{x}}{\log{\log{x}}} \left(\log{\sqrt{\frac{q}{p}}} + \left(\frac{\log{\log{\log{x}}}}{\log{\log{x}}}\right)\right) \right)} \,,$$ where the supremum is taken over all non-zero Dirichlet polynomials of the form $D(s)=\sum_{n \leq x}{a_{n} n^{-s}}$. An aplication is given to the study of multipliers between Hardy spaces of Dirichlet series.Comment: 11 page

Hp\mathcal{H}_p-theory of general Dirichlet series

Inspired by results of Bayart on ordinary Dirichlet series $\sum a_n n^{-s}$, the main purpose of this article is to start an $\mathcal{H}_p$-theory of general Dirichlet series $\sum a_n e^{-\lambda_{n}s}$. Whereas the $\mathcal{H}_p$-theory of ordinary Dirichlet series, in view of an ingenious identification of Bohr, can be seen as a sub-theory of Fourier analysis on the infinite dimensional torus $\mathbb{T}^\infty$, the $\mathcal{H}_p$-theory of general Dirichlet series is build as a sub-theory of Fourier analysis on certain compact abelian groups, including the Bohr compactification $\overline{\mathbb{R}}$ of the reals. Our approach allows to extend various important facts on Hardy spaces of ordinary Dirichlet series to the much wider setting of $\mathcal{H}_p$-spaces of general Dirichlet series

Bohnenblust-Hille inequalities for Lorentz spaces via interpolation

We prove that the Lorentz sequence space $\ell_{\frac{2m}{m+1},1}$ is, in a~precise sense, optimal among all symmetric Banach sequence spaces satisfying a Bohnenblust-Hille type inequality for $m$-linear forms or $m$-homogeneous polynomials on $\mathbb{C}^n$. Motivated by this result we develop methods for dealing with subtle Bohnenblust-Hille type inequalities in the setting of Lorentz spaces. Based on an interpolation approach and the Blei-Fournier inequalities involving mixed type spaces, we prove multilinear and polynomial Bohnenblust-Hille type inequalities in Lorentz spaces with subpolynomial and subexponential constants. Improving a remarkable result of Balasubramanian-Calado-Queff\'elec, we show an application to the theory of Dirichlet series

Non-symmetric polarization

Let $P$ be an $m$-homogeneous polynomial in $n$-complex variables $x_1, \dotsc, x_n$. Clearly, $P$ has a unique representation in the form \begin{equation*} P(x)= \sum_{1 \leq j_1 \leq \dotsc \leq j_m \leq n} c_{(j_1, \dotsc, j_m)} \, x_{j_1} \dotsb x_{j_m} \,, \end{equation*} and the $m$"~form \begin{equation*} L_P(x^{(1)}, \dotsc, x^{(m)})= \sum_{1 \leq j_1 \leq \dotsc \leq j_m \leq n} c_{(j_1, \dotsc, j_m)} \, x^{(1)}_{j_1} \dotsb x^{(m)}_{j_m} \end{equation*} satisfies $L_P(x,\dotsc, x) = P(x)$ for every $x\in\mathbb{C}^n$. We show that, although $L_P$ in general is non-symmetric, for a large class of reasonable norms $\lVert \cdot \rVert$ on $\mathbb{C}^n$ the norm of $L_P$ on $(\mathbb{C}^n, \lVert \cdot \rVert )^m$ up to a logarithmic term $(c \log n)^{m^2}$ can be estimated by the norm of $P$ on $(\mathbb{C}^n, \lVert \cdot \rVert )$; here $c \ge 1$ denotes a universal constant. Moreover, for the $\ell_p$"~norms $\lVert \cdot \rVert_p$, $1 \leq p < 2$ the logarithmic term in the number $n$ of variables is even superfluous