Search

In this paper we consider generalized square function norms of holomorphic functions with values in a Banach space. One of the main results is a characterization of embeddings of the form \[L^p(X)\subseteq \gamma(X) \subseteq L^q(X),\] in terms of the type $p$ and cotype $q$ for the Banach space $X$. As an application we prove $L^p$-estimates for vector-valued Littlewood-Paley-Stein $g$-functions and derive an embedding result for real and complex interpolation spaces under type and cotype conditions.

Summary. Geometric structure of Cesàro function spaces Ces_p(I), where I = [0, 1] and [0,\infty), is investigated. Among other matters we present a description of their dual spaces, characterize the sets of all q such that Ces_p[0, 1] contains isomorphic and complemented copies of lq-spaces, show that Cesàro function spaces fail the fixed point property, give a description of subspaces generated by Rademacher functions in spaces Ces_p[0, 1]. Validerad; 2015; Nivå 1; 20141125 (andbra)

We investigate compact operators between approximation spaces, paying special attention to the limit case. Applications are given to embeddings between Besov spaces.

We characterize the associate spaces of weighted Lorentz spaces GΓ(p,m,w) and present some applications of this result including necessary and sufficient conditions for a Sobolev-type embedding into L∞.

We prove a Calderon–Zygmund type estimate which can be applied to sharpen known regularity results on spherical means, Fourier integral operators, generalized Radon transforms and singular oscillatory integrals. The main theme in this paper is to strengthen various sharp L–Sobolev regularity results for integral operators. To illustrate this we consider the example of spherical means. Let σ denote surface measure on the unit sphere. Since |σ(ξ)| 6 C(1 + |ξ|) d−1 2 the convolution operator f 7→ f ∗ σ maps L to the Sobolev space L(d−1)/2. By complex interpolation with an L–BMO–estimate, Fefferman and Stein [4] proved that the operator maps L to Lp(d−1)/p for 2 1 and in a neighborhood of the origin for k = 0; we assume that ∑∞ k=0 Πkf = f for all Schwartz functions. It is well known, and immediate from Littlewood–Paley theory and embeddings for sequence spaces, that L ⊂ F p 0,p ≡ B p 0,p, 2 6 p < ∞ 1991 Mathematics Subject Classification. 42B20, 42B35, 35S30.

We characterize those classes $\mathcal{C}$ of separable Banach spaces for which there exists a separable Banach space $Y$ not containing $\ell_1$ and such that every space in the class $\mathcal{C}$ is a quotient of $Y$., 23 pages, no figures

We create a new family of Banach spaces, the James‐Schreier spaces, by amalgamating two important classical Banach spaces: James’ quasi-reflexive Banach space on the one hand and Schreier’s Banach space giving a counterexample to the Banach‐Saks property on the other. We then investigate the properties of these James‐Schreier spaces, paying particular attention to how key properties of their ‘ancestors’ (that is, the James space and the Schreier space) are expressed in them. Our main results include that each James‐Schreier space is c0-saturated and that no James‐Schreier space embeds in a Banach space with an unconditional basis. 2010 Mathematics Subject Classification: Primary 46B45; Secondary 46B03, 47B37.

The concept of lushness was introduced recently as a Banach space property, which ensures that the space has numerical index 1. We prove that for Asplund spaces lushness is actually equivalent to numerical index 1. We prove that every separable Banach space containing an isomorphic copy of c0 can be renormed equivalently to be lush, and thus to have numerical index 1. The rest of the paper is devoted to the study of lushness just as a property of Banach spaces. We prove that lushness is separably determined, is stable under ultraproducts, and we characterize those spaces of the form X = (R, ‖ · ‖) with absolute norm such that X-sum preserves lushness of summands, showing that this property is equivalent to lushness of X.

We study atomic decompositions and their relationship with duality and reflexivity of Banach spaces. To this end, we extend the concepts of “shrinking” and “boundedly complete” Schauder basis to the atomic decomposition framework. This allows us to answer a basic duality question: when an atomic decomposition for a Banach space generates, by duality, an atomic decomposition for its dual space. We also characterize the reflexivity of a Banach space in terms of properties of its atomic decompositions.

By a ball-coveringB of a Banach space X, we mean a collection of open balls o the origin in X and whose union contains the unit sphere of X; a ball-covering B is called minimal if its cardinalityB # is smallest among all ball-coverings of X. This article, through establishing a characterization for existence of a ball-covering in Banach spaces, shows that for every n 2 N with k n there exists an n-dimensional space admitting a minimal ball-covering of n + k balls. As an application, we give a new char- acterization of superreflexive spaces in terms of ball-coverings. Finally, we show that ev- ery infinite-dimensional Banach space admits an equivalent norm such that there is an infinite-dimensional quotient space possessing a countable ball-covering. 1. Introduction. The study of geometric and topological properties of unit balls of normed spaces plays a central rule in the geometry of Banach spaces. Almost all properties of Banach spaces, such as convexity, smooth

We study the boundary structure for w -compact subsets of dual Banach spaces. To be more precise, for a Banach space X, 0 < " < 1 and a subset T of the dual space X such that S fB(t;") : t 2 Tg contains a James boundary for BX we study dierent

We consider the problem of isometric embedding of metric spaces into Banach spaces, and introduce and study the remarkable class of so-called linearly rigid metric spaces: these are the spaces that admit a unique, up to isometry, linearly dense isometric embedding into a Banach space. The first nontrivial example of such a space was given by R. Holmes; he proved that the universal Urysohn space has this property. We give a criterion of linear rigidity of a metric space, which allows us to give a simple proof of the linear rigidity of the Urysohn space and some other metric spaces. Various properties of linearly rigid spaces and related spaces are considered.

In this paper we study the non-autonomous stochastic Cauchy problem on a real Banach space E, dU(t) = A(t)U(t) dt+B(t) dWH(t), t ∈ [0, T ], U(0) = u0. Here,WH is a cylindrical Brownian motion on a real separable Hilbert spaceH, (B(t))t∈[0,T ] are closed and densely defined operators from a constant domain D(B) ⊂ H into E, (A(t))t∈[0,T ] denotes the generator of an evolution family on E, and u0 ∈ E. In the first part, we study existence of weak and mild solutions by methods of van Neerven and Weis. Then, we use a well-known factorisation method in the setting of evolution families to obtain time regularity of the solution. In the second part, we consider the parabolic case in the setting of Acquistapace and Terreni. By means of a factorisation method in the spirit of Da Prato, Kwapien, and Zabczyk we obtain space-time regularity results in the context of parabolic evolution families on Banach spaces. Afterwards, we apply this theory to several examples. In the last part, relying on recent results of Dettweiler, van Neerven, and Weis, we prove a maximal regularity result where the A(t) are as in the setting of Kato and Tanabe.

We generalize the theory of Wiener amalgam spaces on locally compact groups to quasi-Banach spaces. As a main result we provide convolution relations for such spaces. Also we weaken the technical assumption that the global component is invariant under right translations, which is even new for the classical Banach space case. To illustrate our theory we discuss in detail an example on the $ax+b$ group., Comment: 20 pages

We de ne a Sobolev space by means of a generalized Poincare inequality and relate it to a corresponding space based on upper gradients. 2000 Mathematics Subject Classi cation: Primary 46E35, Secondary 46E30, 26D10

We study representations of Banach algebras on reflexive Banach spaces. Algebras which admit such representations which are bounded below seem to be a good generalisation of Arens regular Banach algebras; this class includes dual Banach algebras as defined by Runde, but also all group algebras, and all discrete (weakly cancellative) semigroup algebras. Such algebras also behave in a similar way to C$^*$- and W$^*$-algebras; we show that interpolation space techniques can be used in the place of GNS type arguments. We define a notion of injectivity for dual Banach algebras, and show that this is equivalent to Connes-amenability. We conclude by looking at the problem of defining a well-behaved tensor product for dual Banach algebras., Comment: 40 pages; Update corrects some mathematics, and merges two sections to make for easier reading