The classical representation of random variables as the Itô integral of nonanticipative integrands is extended to include Banach space valued random variables on an abstract Wiener space equipped with a filtration induced by a resolution of the identity on the Cameron–Martin space. The Itô integral is replaced in this case by an extension of the divergence to random operators, and the operators...

Two versions of Rubio de Francia’s extrapolation theorem for multivariable operators of functions are obtained. One version assumes an initial estimate with different weights in each space and implies boundedness on all products of Lebesgue spaces. Another version assumes an initial estimate with the same weight but yields boundedness on a product of Lebesgue spaces whose indices lie on a line....

In this paper it is shown how the vector-valued Laplace transform can be used to obtain existence and uniqueness results for exponentially bounded solutions of abstract linear evolution equations in Banach spaces. The method allows to consider problems for which the characteristic operator is, like many sums, products or limits of closed operators, not necessarily closed or closable.

Recent reverses for the discrete generalised triangle inequality and its continuous version for vector-valued integrals in Banach spaces are surveyed. New results are also obtained. Particular instances of interest in Hilbert spaces and for complex numbers and functions are pointed out as well.

we establish a sharp maximal function estimate for some vector-valued multilinear singular integral operators. as an application, we obtain the $(l^p, l^q)$-norm inequality for vector-valued multilinear operators.

Abstract The concept of Rademacher type p (1 ≤ p ≤ 2) plays an important role in the local theory of Banach spaces. In [3] Mascioni considers a weakening of this concept and shows that for a Banach space X weak Rademacher type p implies Rademacher type r for all r < p. As with Rademacher type p and weak Rademacher type p, we introduce the concept of Haar type p and weak Haar type p by replacing...

in this paper, we introduce the idea of relative order and type of entire functions represented by banach valued dirichlet series of two complex variables to generalize some earlier results.proving some preliminary theorems on the relative order, we obtain sum and product theorems and we show that the relative order of an entire function represented by dirichlet series is the same as that of it...

2. Theorem. Let £ be a reflexive B-space and let (S, 2, m) be a a-finite measure space. Let there be defined on S a strongly measurable function Ts with values in the B-space B(H) of bounded linear operators on H. Suppose that || 7\|| ^=1 for all sES. Let h be a measure-preserving transformation (m.p.t.) in (S, 2, m). Then for each XELi(S, £) there is an XELi(S, X) such that limn<„ «_1E"=i T*Th...