Laplace Transforms and Integrated, Regularized Semigroups in Locally Convex Spaces

We study the metric and w^* -differentiability of pointwise Lipschitz mappings. First, we prove several theorems about metric and w^* -differentiability of pointwise Lipschitz mappings between \mathbb{R}^n and a Banach space X (which extend results due to Ambrosio, Kirchheim and others), then apply these to functions satisfying the spherical Rado–Reichelderfer condition, and to absolutely continuous functions of several variables with values in a Banach space. We also establish the area formula for pointwise Lipschitz functions, and for (n,\lambda) -absolutely continuous functions with values in Banach spaces. In the second part of this paper, we prove two theorems concerning metric and w^* -differentiability of pointwise Lipschitz mappings f:X\mapsto Y where X,Y are Banach spaces with X being separable (resp. X separable and Y=G^* with G separable).

UDC 517.5 Let ( 𝕏 , ‖ . ‖ 𝕏 ) denote a complex Banach space and L ( 𝕏 ) = B C ( ℝ → 𝕏 ) be the set of all 𝕏 -valued bounded continuous functions f : ℝ → 𝕏 . For f ∈ L ( 𝕏 ) we define ‖ f ‖ L ( 𝕏 ) = sup { ‖ f ( x ) ‖ 𝕏 : x ∈ ℝ } . Then ( L ( 𝕏 ) , ‖ . ‖ L ( 𝕏 ) ) itself is a Banach space. The Beurling spectrum S p e c ( f ) of a function f ∈ L ( 𝕏 ) is defined by S p e c ( f ) = { ζ ∈ ℝ : ∀ ϵ > 0 ∃ φ ∈ 𝒮 ( ℝ ) : supp φ ^ ⊂ ( ζ - ϵ , ζ + ϵ ) , φ * f ≢ 0 } . } . We obtain the following Paley-Wiener type theorem for functions with values in Banach spaces: Let f ∈ L ( 𝕏 ) and K be an arbitrary compact set in ℝ . Then Spec ( f ) ⊂ K if and only if for any τ > 0 there exists a constant C τ < ∞ such that < b r > ‖ P ( D ) f ‖ L ( 𝕏 ) ≤ C τ < b r > ‖ f ‖ L ( 𝕏 ) sup x ∈ K ( τ ) | P ( x ) | < b r > for all polynomials with complex coefficients P ( x ) , where the differential operator P ( D ) is obtained from P ( x ) by substituting x → - i d ⅆ x , d ⅆ x is the usual derivative in L ( 𝕏 ) and K ( τ ) is the τ -neighborhood in ℂ of K . Moreover, Paley-Wiener type theorem for integral operators and one for some special compacts K are also given.

Using traditional methods it is possible to prove the Ito formula in a Hilbert space and some Banach spaces with special geometrical properties. The class of such Banach spaces is very narrow-they are subclass of reflexive Banach spaces. Using the definition of a generalized stochastic integral, early we proved the Ito formula in an arbitrary Banach space for the case, when as initial Ito process was the Wiener process. For an arbitrary Banach space and an arbitrary Ito process it is impossible to find the sequence of corresponding step functions with the desired convergence. We consider the space of generalized random processes, introduce general Ito process there and prove in it the Ito formula. Afterward, from the main Ito process in a Banach space we receive the generalized Ito process in the space of generalized random processes and we get the Ito formula in this space. Then we check decompasibilility of the members of the received equality and as they turn out Banach space valued, we get the Ito formula in an arbitrary Banach space. We implemented this approach when the stochastic integral in the Ito process was taken from a Banach space valued non-anticipating random process by the one dimensional Wiener process. In this paper we consider the case, when the stochastic integral is taken from an operator- valued non-anticipating random process by the Wiener process with values in a Banach space.

Here we research the univariate multi-composite sigmoid activated quantitative approximation of Banach space valued continuous functions on a compact interval or all the real line by quasi-interpolation Banach space valued multi-composite sigmoid activated neural network operators. We perform also the related Banach space valued multi-composite sigmoid activated fractional approximation. These multi-composite sigmoid acti-vated approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its Banach space valued high order derivative or fractional derivatives. Our operators are deÖned by using a multi-composite density function induced by a gen- eral multi-composite sigmoid function. The approximations are pointwise and with respect to the uniform norm. The related Banach space valued feed-forward neural networks are with one hidden layer. We Önish with a convergence analysis.

We derive moment and tail estimates for Gaussian chaoses of arbitrary order with values in Banach spaces. We formulate a conjecture regarding two-sided estimates and show that it holds in a certain class of Banach spaces including $L_{q}$ spaces. As a corollary we obtain two-sided bounds for moments of chaoses with values in $L_{q}$ spaces based on exponential random variables.

The purpose of this paper is twofold: first, to extend the definition of the stochastic integral for processes with values in Banach spaces; and second, to define the stochastic integral as a genuine integral, with respect to a measure, that is, to provide a general integration theory for vector measures, which, when applied to stochastic processes, yields the stochastic integral along with all its properties. For the reader interested only in scalar stochastic integration, our approach should still be of interest, since it sheds new light on the stochastic integral, enlarges the class of integrable processes and presents new convergence theorems involving the stochastic integral.

In the present paper we consider the transition semigroup Pt related to some stochastic reaction-diAusion equations with the non- linear term f having polynomial growth and satisfying some dis- sipativity conditions. We are proving that it has a regularizing eAect in the Banach space of continuous functions COOU, where O R d is a bounded open set. In L 2 OOU the only result proved is the strong Feller property, for da 1. Here we are able to prove that if f 2 C 1 ORU and d 3, then Ptu2 C 1 b OCOOUU for any u2 BbOCOOUU and t > 0. An important application is to the study of the ergodic properties of the system. These results are also of interest for some problem in sto- chastic control.

On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions

This paper is concerned with a version of the Lebesgue dominated convergence theorem (DCT) which has been stated for the Kurzweil–Stieltjes integral of real functions. Our objective in this work is to analyze the extension of this result to include vector functions with values in Banach spaces. We establish that the mentioned convergence theorem for the Kurzweil–Stieltjes integral can be formulated in weaker versions for reflexive and separable Banach spaces, and spaces having the Schur property, nonetheless it is not verified in the general case.

We formulate standard and multilevel Monte Carlo methods for the kth moment Mεk[ξ] of a Banach space valued random variable ξ:Ω→E, interpreted as an element of the k-fold injective tensor product space ⊗εkE. For the standard Monte Carlo estimator of Mεk[ξ], we prove the k-independent convergence rate 1−1p in the Lq(Ω;⊗εkE)-norm, provided that (i) ξ∈Lkq(Ω;E) and (ii) q∈[p,∞), where p∈[1,2] is the Rademacher type of E. By using the fact that Rademacher averages are dominated by Gaussian sums combined with a version of Slepian's inequality for Gaussian processes due to Fernique, we moreover derive corresponding results for multilevel Monte Carlo methods, including a rigorous error estimate in the Lq(Ω;⊗εkE)-norm and the optimization of the computational cost for a given accuracy. Whenever the type of the Banach space E is p=2, our findings coincide with known results for Hilbert space valued random variables.We illustrate the abstract results by three model problems: second-order elliptic PDEs with random forcing or random coefficient, and stochastic evolution equations. In these cases, the solution processes naturally take values in non-Hilbertian Banach spaces. Further applications, where physical modeling constraints impose a setting in Banach spaces of type p<2, are indicated.

In a recent paper, F. Bombal and P. Cembranos showed that if E is a Banach space such that E* is separable, then C(Q, E), the Banach space of continuous functions from a compact Hausdorff space Q to E, has the Dieudonne property. They asked whether or not the result is still true if one only assumes that E does not contain a copy of l1. In this paper we give a positive answer to their question. As a corollary we show that if E is a subspace of an order continuous Banach lattice, then E has the Dieudonne property if and only if C(Q, E) has the same property. If E is a Banach space and Q is a compact Hausdorff space, then C(Q, E) will stand for the Banach space of the E-valued continuous functions on Q under the supremum norm. A Banach space E is said to have the Dieudonne property if for every Banach space F, any bounded linear operator T: E -> F that transforms weakly Cauchy sequences into weakly convergent sequences is weakly compact. In [3] F. Bombal and P. Cembranos showed that if E is a Banach space such that E* is separable, then C(Q, E) has the Dieudonne property and they asked whether the same result is true when replacing the assumption that E* is separable by supposing only that 11 does not embed in E. In this paper we give a positive answer to their question. Recall that a topological space (X, -y) is said to be Polish if it is homeomorphic to a separable complete metric space and it is said to be analytic if it is the continuous image of a Polish space. A subset A of a topological space (X, y) is said to be coanalytic if its complement (X\A, y) is analytic. Finally A is said to be PCA if it is the continuous image of a coanalytic space. The notations and terminology used and not defined can be found in [5, 8, or 10]. In the proof of Lemma 3 we need the following two results. THEOREM 1 (M. SREBRNY [9]). Let X and Y be two analytic spaces and let F be a multivalued function from X to the subsets of Y, such that its graph is PCA and for which one can prove that for every x c X, F(x) :8 0 using only the axioms of ZFC. Then there exists a universally measurable map f: X -> Y such that f (x) c F(x) for every x E X. THEOREM 2 (I. ASSANI [1, 2]). Let E be a separable Banach space. The set of weakly Cauchy sequences is a coanalytic subset of EN. Received by the editors January 5, 1985. 1980 Mathematics Subject Classification. Primary 46G10, 46B22.

This paper gives strong and weak laws of large numbers for random elements with values in Banach spaces of type $(F,F_1 )$. The known laws of large numbers in Banach spaces of stable and Rademacher type p are special cases of these results. Characterizations of spaces of type $(F,F_1 )$ are given in terms of these laws.

We prove some results on weakly almost square Banach spaces and their relatives. On the one hand, we discuss weak almost squareness in the setting of Banach function spaces. More precisely, let $(\Omega,\Sigma)$ be a measurable space, let E be a Banach lattice and let $\nu:\Sigma \to E^+$ be a non-atomic countably additive measure having relatively norm compact range. Then the space $L_1(\nu)$ is weakly almost square. This result applies to some abstract Cesàro function spaces. Similar arguments show that the Lebesgue–Bochner space $L_1(\mu,Y)$ is weakly almost square for any Banach space Y and for any non-atomic finite measure µ. On the other hand, we make some progress on the open question of whether there exists a locally almost square Banach space, which fails the diameter two property. In this line, we prove that if X is any Banach space containing a complemented isomorphic copy of c0, then for every $0 \lt \varepsilon \lt 1$, there exists an equivalent norm $|\cdot|$ on X satisfying the following: (i) every slice of the unit ball $B_{(X,|\cdot|)}$ has diameter 2; (ii) $B_{(X,|\cdot|)}$ contains non-empty relatively weakly open subsets of arbitrarily small diameter and (iii) $(X,|\cdot|)$ is (r, s)-SQ for all $0 \lt r,s \lt \frac{1-\varepsilon}{1+\varepsilon}$.

Continuous framings for Banach spaces

Function spaces and its various applications are subjects of investigation in many research centers and we are glad that this special issue has received the remarkable attention by researchers in these areas. We have received papers from wide range of topics of function spaces such as interpolation methods for stochastic processes spaces, vector optimization problems in Banach spaces, Marcinkiewicz integral operator in Morrey spaces, Hardy’s operator in the variable exponent Lebesgue spaces, and differential equations of weakly parabolic type in Banach spaces. A number of papers are devoted to investigating different kinds of problems in Hardy spaces or Hardy subspaces, for example, Calderon-Zygumnd singular integral operators in Hardy spaces, weighted Hardy spaces related to the Schrodinger operator, and composition operator in hyperbolic Bloch-type spaces. Geometric problems such as modulus of convexity in some Banach sequence spaces, p-uniform convexity, and quniform smoothness in C spaces are also covered by this issue.