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We prove various extensions of the Coifman–Rubio de Francia–Semmes multiplier theorem to operator-valued multipliers on Banach function spaces. Our results involve a new boundedness condition on sets of operators which we call ℓ r ( ℓ s ) {\ell ^{r}(\ell ^{s})} -boundedness, which implies R \mathcal {R} -boundedness in many cases. The proofs are based on new Littlewood–Paley–Rubio de Francia-type estimates in Banach function spaces which were recently obtained by the authors.

In this work we study the following three space problem for operator spaces: if X is an operator space with base space isomorphic to a Hilbert space and X contains a completely isomorphic copy of the operator Hilbert space OH with respective quotient also completely isomorphic to OH, must X be completely isomorphic to OH? This problem leads us to the study of short exact sequences of operator spaces, more specifically those induced by complex interpolation, and their splitting. We show that the answer to the three space problem is negative, giving two different solutions.

Optimal target spaces are exhibited in arbitrary-order Sobolev type embeddings for traces of n n -dimensional functions on lower dimensional subspaces. Sobolev spaces built upon any rearrangement-invariant norm are allowed. A key step in our approach consists of showing that any trace embedding can be reduced to a one-dimensional inequality for a Hardy type operator depending only on n n and on the dimension of the relevant subspace. This can be regarded as an analogue for trace embeddings of a well-known symmetrization principle for first-order Sobolev embeddings for compactly supported functions. The stability of the optimal target space under iterations of Sobolev trace embeddings is also established and is part of the proof of our reduction principle. As a consequence, we derive new trace embeddings, with improved (optimal) target spaces, for classical Sobolev, Lorentz-Sobolev and Orlicz-Sobolev spaces.

This paper is inspired by the work of Nagel and Stein in which the L p L^p ( 1 > p > ∞ ) (1>p>\infty ) theory has been developed in the setting of the product Carnot-Carathéodory spaces M ~ = M 1 × ⋯ × M n \widetilde {M}=M_1\times \cdots \times M_n formed by vector fields satisfying Hörmander’s finite rank condition. The main purpose of this paper is to provide a unified approach to develop the multiparameter Hardy space theory on product spaces of homogeneous type. This theory includes the product Hardy space, its dual, the product B M O BMO space, the boundedness of singular integral operators and Calderón-Zygmund decomposition and interpolation of operators. As a consequence, we obtain the endpoint estimates for those singular integral operators considered by Nagel and Stein (2004). In fact, we will develop most of our theory in the framework of product spaces of homogeneous type which only satisfy the doubling condition and some regularity assumption on the metric. All of our results are established by introducing certain Banach spaces of test functions and distributions, developing discrete Calderón identity and discrete Littlewood-Paley-Stein theory. Our methods do not rely on the Journé-type covering lemma which was the main tool to prove the boundedness of singular integrals on the classical product Hardy spaces.

This paper is an exploration of two ideas in the study of closed classes: the A-complexity of a space X and the notion of good spaces (spaces A for which C(A) = C(A)). A variety of formulae for the computation of complexity are given, along with some calculations. Good spaces are characterized in terms of the functors CW A and P A . The main result is a countable upper bound for ΣA-complexity when A is a good space.

A Banach space X has the approximation property if and only if every compact set in X is in the range of a one-to-one bounded linear operator from a space that has a Schauder basis. Characterizations are given for L p spaces and quotients of L p spaces in terms of covering compact sets in X by operator ranges from L p spaces. A Banach space X is a L 1 space if and only if every compact set in X is contained in the closed convex symmetric hull of a basic sequence which converges to zero.

We extend lemmas by Bourgain-Brezis-Mironescu (2001), and Maz'ya-Shaposhnikova (2002), on limits of Sobolev spaces, to the setting of interpolation scales. This is achieved by means of establishing the continuity of real and complex interpolation scales at the end points. A connection to extrapolation theory is developed, and a new application to limits of Sobolev scales is obtained. We also give a new approach to the problem of how to recognize constant functions via Sobolev conditions.

The Banach space $E$ has the weakly compact approximation property (W.A.P. for short) if there is a constant $C < \infty$ so that for any weakly compact set $D \subset E$ and $\epsilon > 0$ there is a weakly compact operator $V: E \to E$ satisfying $\sup_{x\in D} || x - Vx || < \epsilon$ and $|| V|| \leq C$. We give several examples of Banach spaces both with and without this approximation property. Our main results demonstrate that the James-type spaces from a general class of quasi-reflexive spaces (which contains the classical James' space $J$) have the W.A.P, but that James' tree space $JT$ fails to have the W.A.P. It is also shown that the dual $J^*$ has the W.A.P. It follows that the Banach algebras $W(J)$ and $W(J^*)$, consisting of the weakly compact operators, have bounded left approximate identities. Among the other results we obtain a concrete Banach space $Y$ so that $Y$ fails to have the W.A.P., but $Y$ has this approximation property without the uniform bound $C$., Comment: 39 pages, plain tex document

We study the action of composition operators on Sobolev spaces of analytic functions having fractional derivatives in some weighted Bergman space or Hardy space on the unit disk. Criteria for when such operators are bounded or compact are given. In particular, we find the precise range of orders of fractional derivatives for which all composition operators are bounded on such spaces. Sharp results about boundedness and compactness of a composition operator are also given when the inducing map is polygonal.

This paper is devoted to the analysis of function spaces modeled on Besov spaces and their applications to non-linear partial differential equations, with emphasis on the incompressible, isotropic Navier-Stokes system and semilinear heat equations. Specifically, we consider the class, introduced by Hideo Kozono and Masao Yamazaki, of Besov spaces based on Morrey spaces, which we call Besov-Morrey or BM spaces. We obtain equivalent representations in terms of the Weierstrass semigroup and wavelets, and various embeddings in classical spaces. We then establish pseudo-differential and para-differential estimates. Our results cover non-regular and exotic symbols. Although the heat semigroup is not strongly continuous on Morrey spaces, we show that its action defines an equivalent norm. In particular, homogeneous BM spaces belong to a larger class constructed by Grzegorz Karch to analyze scaling in parabolic equations. We compare Karch's results with those of Kozono and Yamazaki and generalize them by obtaining short-time existence and uniqueness of solutions for arbitrary data with subcritical regularity. We exploit pseudo-differential calculus to extend the analysis to compact, smooth, boundaryless, Riemannian manifolds. BM spaces are defined by means of partitions of unity and coordinate patches, and intrinsically in terms of functions of the Laplace operator.

An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree T \mathcal {T} of a certain type on a space X X is presumed to have a branch with some property. It is shown that then X X can be embedded into a space with an FDD ( E i ) (E_i) so that all normalized sequences in X X which are almost a skipped blocking of ( E i ) (E_i) have that property. As an application of our work we prove that if X X is a separable reflexive Banach space and for some 1 > p > ∞ 1>p>\infty and C > ∞ C>\infty every weakly null tree T \mathcal {T} on the sphere of X X has a branch C C -equivalent to the unit vector basis of ℓ p \ell _p , then for all ε > 0 \varepsilon >0 , there exists a subspace of X X having finite codimension which C 2 + ε C^2+\varepsilon embeds into the ℓ p \ell _p sum of finite dimensional spaces.

In this paper we define a distinguished boundary for the complex crowns Ξ ⊆ G C /K C of non-compact Riemannian symmetric spaces G/K. The basic result is that affine symmetric spaces of G can appear as a component of this boundary if and only if they are non-compactly causal symmetric spaces.

Wavelets are generated from refinable functions by using multiresolution anlalysis. In this paper we investigate the srloothness properties of multivariate refinable functionis in Sobolev spaces. We characterize the optinlal smoothness of a multivariate refinable function in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite dimensional invariant subspace. Several examples are provided to illustrate the general theory.

For Banach spaces X X and Y Y and a bounded linear operator T : X → Y T:X \rightarrow Y we let ρ ( T ) := inf c \rho (T):=\inf c such that \[ ( A V θ l = ± 1 ‖ ∑ l = 1 ∞ θ l ( ∑ k = τ l − 1 + 1 τ l h k T x k ) ‖ L 2 Y 2 ) 1 2 ≤ c ‖ ∑ k = 1 ∞ h k x k ‖ L 2 X \left ( AV_{\theta _l = \pm 1} \left \|\sum \limits _{l=1}^\infty \theta _l \left ( \sum \limits _{k=\tau _{l-1}+1}^{\tau _l} h_k T x_k \right )\right \|_{L_2^Y}^2 \right )^{\frac {1}{2}} \le c \left \| \sum \limits _{k=1}^\infty h_k x_k \right \| _{L_2^X} \] for all finitely supported ( x k ) k = 1 ∞ ⊂ X (x_k)_{k=1}^\infty \subset X and all 0 = τ 0 > τ 1 > ⋯ 0 = \tau _0 > \tau _1 > \cdots , where ( h k ) k = 1 ∞ ⊂ L 1 [ 0 , 1 ) (h_k)_{k=1}^\infty \subset L_1[0,1) is the sequence of Haar functions. We construct an operator T : X → X T:X \rightarrow X , where X X is superreflexive and of type 2, with ρ ( T ) > ∞ \rho (T)>\infty such that there is no constant c > 0 c>0 with \[ sup θ k = ± 1 ‖ ∑ k = 1 ∞ θ k h k T x k ‖ L 2 X ≤ c ‖ ∑ k = 1 ∞ h k x k ‖ L 2 X . \sup _{\theta _k = \pm 1} \left \| \sum \limits _{k=1}^\infty \theta _k h_k T x_k \right \| _{L_2^X} \le c \left \| \sum \limits _{k=1}^\infty h_k x_k \right \| _{L_2^X}. \] In particular it turns out that the decoupling constants ρ ( I X ) \rho (I_X) , where I X I_X is the identity of a Banach space X X , fail to be equivalent up to absolute multiplicative constants to the usual UMD \operatorname {UMD} –constants. As a by-product we extend the characterization of the non–superreflexive Banach spaces by the finite tree property using lower 2–estimates of sums of martingale differences.

We investigate the stability of Fredholm properties on interpolation scales of quasi-Banach spaces. This analysis is motivated by problems arising in PDE’s and several applications are presented.

We consider a class of elliptic and parabolic differential operators with unbounded coefficients in R n \mathbb R^n , and we study the properties of the realization of such operators in suitable weighted L 2 L^2 spaces.

Two examples of asymptotic ℓ 1 \ell _{1} Banach spaces are given. The first, X u X_{u} , has an unconditional basis and is arbitrarily distortable. The second, X X , does not contain any unconditional basic sequence. Both are spaces of the type of Tsirelson’s.

Let G G be a semigroup and a topological space. Let X X be an Abelian topological group. The right differences △ h φ \triangle _{h} \varphi of a function φ : G → X \varphi : G \to X are defined by △ h φ ( t ) = φ ( t h ) − φ ( t ) \triangle _{h}\varphi (t) = \varphi (th) - \varphi (t) for h , t ∈ G h,t \in G . Let △ h φ \triangle _{h} \varphi be continuous at the identity e e of G G for all h h in a neighbourhood U U of e e . We give conditions on X X or range φ \varphi under which φ \varphi is continuous for any topological space G G . We also seek conditions on G G under which we conclude that φ \varphi is continuous at e e for arbitrary X X . This led us to introduce new classes of semigroups containing all complete metric and locally countably compact quasitopological groups. In this paper we study these classes and explore their relation with Namioka spaces.

A probabilistic proof is given to identify the complex interpolation space of H 1 ( T ) {H^1}(\mathbb {T}) and H ∞ ( T ) {H^\infty }(\mathbb {T}) as H p ( T ) {H^p}(\mathbb {T}) .

Analytic capacities are set functions defined on the plane which may be used in the study of removable singularities, boundary smoothness and approximation of analytic functions belonging to some function space. The symmetric concrete Banach spaces form a class of function spaces that includes most spaces usually studied. The Beurling transform is a certain singular integral operator that has proved useful in analytic function theory. It is shown that the analytic capacity associated to each Beurling-invariant symmetric concrete Banach space behaves reasonably under affine transformation of the plane. It is not known how general analytic capacities behave under affine maps.

Real and complex interpolation methods, when applied to the couple ( H p 0 ( E 0 ; w 0 ) , H p 1 ( E 1 ; w 1 ) ) ({H^{{p_0}}}({E_0};{w_0}),{H^{{p_1}}}({E_1};{w_1})) , give what is expected if E 0 {E_0} and E 1 {E_1} are quasi-Banach lattices of measurable functions satisfying certain mild conditions and if log ⁡ ( w 0 1 / p 0 w 1 − 1 / p 1 ) ∈ BMO ( w 0 , w 1 \log (w_0^{1/{p_0}}w_1^{ - 1/{p_1}}) \in {\text {BMO}}\,({w_0},{w_1} being weights on the unit circle). The last condition is in fact necessary. (It is expected, of course, that the resulting spaces coincide with the subspaces of analytic functions in the corresponding interpolation spaces for the couple ( L p 0 ( E 0 ; w 0 ) , L p 1 ( E 1 ; w 1 ) ) . ) ({L^{{p_0}}}({E_0};{w_0}),{L^{{p_1}}}({E_1};{w_1})).)

This paper addresses some questions which arise naturally in the theory of nests of subspaces in Banach space. The order topology on the index set of a nest is discussed, as well as the method of spatial indexing by a vector; sufficient geometric conditions for the existence of such a vector are found. It is then shown that a continuous nest exists in any Banach space. Applications and examples follow; in particular, an extension of the Volterra nest in L ∞ [ 0 , 1 ] {L^\infty }[ {0,1} ] to a continuous one, a continuous nest in a Banach space having no two elements isomorphic to one another, and a characterization of separable L p {\mathcal {L}_p} -spaces in terms of nests.

Generalized second-order derivatives introduced by Rockafellar in finite-dimensional spaces are extended to convex functions in reflexive Banach spaces. Parallel results are shown in the infinite-dimensional case. A result that plays an important role in applications is that the generalized second-order differentiability is preserved under the integral sign.

We continue the study of centralizers on Köthe function spaces and the commutator estimates they generate (see [29]). Our main result is that if X X is a super-reflexive Köthe function space then for every real centralizer Ω \Omega on X X there is a complex interpolation scale of Köthe function spaces through X X inducing Ω \Omega as a derivative, up to equivalence and a scalar multiple. Thus, in a loose sense, all real centralizers can be identified with derivatives of complex interpolation processes. We apply our ideas in an appendix to show, for example, that there is a twisted sum of two Hilbert spaces which fails to be a ( UMD ) ({\text {UMD}}) -space.

We study a spectral sequence converging to the cohomology of the configuration space of n ordered points in a manifold. A chain complex is constructed with homology equal to the E2 term. If the field is the rationals and the manifold is formal then the spectral sequence is shown to collapse. The results are applied to compute the Anderson spectral sequence converging to the cohomology of a function space. 0. INTRODUCTION From the point of view of homotopy theory, a very natural problem is to describe the topology of function spaces. In 1956, R. Thom [13] wrote a seminal paper. In 1972, D. Anderson [1], using the notion of a cosimplicial space introduced by Bousfield and Kan [4], obtained a spectral sequence for the homology of function spaces. In 1969-1970, Gelfand and Fuks [6] wrote a series of papers on the continuous cohomology of the Lie algebra of vector fields on a manifold M. They introduced a pair of spectral sequences for such a study. Based on the results of Gelfand-Fuks Bott and Fuks conjectured that the continuous cohomology of the Lie algebra of vector fields was the usual cohomology of a certain function space, namelyS that of the space of sections of a certain associated bundle to the tangent bundle of M. Haefliger [7] and Bott and Segal [3] in 1978> using entirely different methods, proved the Bott-Fuks conjecture. The method followed by Bott and Segal was to utilize the ideas of Anderson. The cohomology of "configuration spaces' appears in the computation of the El-term of the above spectral sequences. In 1978 F. Cohen and L. Taylor [5] were able to compute the cohomology of the configuration spaces for a certain class of manifolds and applied these results to describe the Gelfand-Fuks cohomology of these manifolds. Also, there has been interest in the configuration spaces as they appear as approximations to certain function spaces (see [2, 9]). In this paper we study the configuration spaces and the function spaces as outlined by Bott-Segal. In both instances the cohomology can be obtained from a Received by the editors October 3, 1987 and in revise(l form, July 1, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 54C35.

Given Banach spaces E E and F F , a Banach space G E F {G_{EF}} is presented in which E E is embedded and which seems a natural space to which extend F F -valued analytic functions. Any F F -valued analytic function defined on a subset U U of E E may be extended to an open neighborhood of U U in G E F {G_{EF}} . This extension generalizes that of Aron and Berner. It is also related to the Arens product in Banach algebras, to the functional calculus for bounded linear operators, and to an old problem of duality in spaces of analytic functions. A characterization of the Aron-Berner extension is given in terms of continuity properties of first-order differentials.

This paper is a continuation of the study of harmonic interpolation families of normed or quasinormed spaces parametrized by points of a domain in C k {{\mathbf {C}}^k} . It is shown, among other things, that each of the following properties holds for all the intermediate quasinormed spaces, if it holds for all given boundary spaces: (1) being a normed space; (2) being a Hilbert space; (3) satisfying the triangle inequality by the r r th power of the quasinorm; (4) being uniformly convex; and (5) being uniformly smooth. As a principal tool, the notion of a harmonic set valued function (a generalization of analytic multifunction) is introduced and studied.

We investigate the connections between the "global" structure of a Banach space (i.e. the existence of certain finite and infinite dimensional decompositions) and the geometrical properties of the closed convex bounded subsets of such a space (i.e. the existence of extremal and other topologically distinguished points). The global structures of various—supposedly pathological— examples of Banach spaces constructed by R. C. James turn out to be more "universal" than expected. For instance James-tree-type (resp. James-matrix-type) decompositions characterize Banach spaces with the Point of Continuity Property (resp. the Radon-Nikodým Property). Moreover, the Convex Point of Continuity Property is stable under the formation of James-infinitely branching tree-type "sums" of infinite dimensional factors. We also give several counterexamples to various questions relating some topological and geometrical concepts in Banach spaces.

We give an estimate relating the stable type p p constant of a Banach space X X with the dimension of the l p n l_p^n -subspaces of X X . Precisely, let C C be this constant and assume 1 > p > 2 1 > p > 2 . We show that, for each ε > 0 , X \varepsilon > 0,X must contain a subspace ( 1 + ε ) (1 + \varepsilon ) -isomorphic to l p k l_p^k , for every k k less than δ ( ε ) C p ′ \delta (\varepsilon ){C^{p’}} where δ ( ε ) > 0 \delta (\varepsilon ) > 0 is a number depending only on p p and ε \varepsilon .

We show that for a measure γ \gamma on a Banach space directional differentiability implies quasi-translation invariance. This result is shown to imply the Cameron-Martin theorem. A second application is given in which γ \gamma is the image of a Gaussian measure under a suitably regular map.

Given a domain Ω ⊂ R N \Omega \subset {R_N} and a Borel function h : R m → R h:\,{R_m} \to R , conditions on h are sought ensuring that for every m-tuple of functions u i {u_i} belonging to the first order Sobolev space W 1 , p ( Ω ) {W^{1,p}}(\Omega ) , the function h ( u 1 ( ⋅ ) , … , u m ( ⋅ ) ) h({u_1}( \cdot ), \ldots ,{u_m}( \cdot )) will belong to a first order Sobolev space W 1 , r ( Ω ) {W^{1,r}}(\Omega ) , 1 ⩽ r ⩽ p > ∞ 1 \leqslant r \leqslant p > \infty .In this paper conditions are found which are both necessary and sufficient in order that h have the above property. This result is based on a characterization obtained here for those Borel functions g : R m × ( R N ) m → R g:\,{R_m} \times {({R_N})_m} \to R satisfying the requirement that for every m-tuple of functions u i ∈ W 1 , p ( Ω ) {u_i} \in {W^{1,p}}(\Omega ) the function g ( u 1 ( ⋅ ) , … , u m ( ⋅ ) , ∇ u 1 ( ⋅ ) , … , ∇ u m ( ⋅ ) ) g({u_1}( \cdot ), \ldots ,{u_m}( \cdot ),\nabla {u_1}( \cdot ), \ldots ,\nabla {u_m}( \cdot )) belongs to L r ( Ω ) {L^r}(\Omega ) . A needed result on the measurability of the set of R k {R_k} -Lebesgue points of a function on R N {R_N} is presented in an appendix.

The present paper is an approach to the local theory of Banach spaces via the ultrapower construction. It includes a detailed study of ultrapowers and their dual spaces as well as a definition of a new notion, the notion of a u-extension of a Banach space. All these tools are used to give a unified definition of many classes of Banach spaces characterized by local properties (such as the L p {\mathcal {L}_p} -spaces). Many examples are given; also, as an application, it is proved that any L p {\mathcal {L}_p} -space, 1 > p > ∞ 1 > p > \infty , has an ultrapower which is isomorphic to an L p {L_p} -space.

We investigate Banach spaces X X such that X X is an M M -ideal in X ∗ ∗ {X^{{\ast }{\ast }}} . Subspaces, quotients and c 0 {c_0} -sums of spaces which are M M -ideals in their biduals are again of this type. A nonreflexive space X X which is an M M -ideal in X ∗ ∗ {X^{{\ast }{\ast }}} contains a copy of c 0 {c_0} . Recently Lima has shown that if K ( X ) K(X) is an M M -ideal in L ( X ) L(X) then X X is an M M -ideal in X ∗ ∗ {X^{{\ast }{\ast }}} . Here we show that if X X is reflexive and K ( X ) K(X) is an M M -ideal in L ( X ) L(X) , then K ( X ) ∗ ∗ K{(X)^{{\ast }{\ast }}} is isometric to L ( X ) L(X) , i.e. K ( X ) K(X) is an M M -ideal in its bidual. Moreover, for real such spaces, we show that K ( X ) K(X) contains a proper M M -ideal if and only if X X or X ∗ {X^{\ast }} contains a proper M M -ideal.

For ϕ ∈ C 2 ( R n ) \phi \in {C^2}({{\mathbf {R}}^n}) with ϕ ( x ) = a | x | 1 + s \phi (x) = a|x{|^{1 + s}} for | x | ⩾ x 0 , a , s > 0 |x| \geqslant {x_0},a,s > 0 , define the measure d μ = exp ⁡ ( − 2 ϕ ) d n x d\mu = \exp ( - 2\phi ){d^n}x on R n {{\mathbf {R}}^n} . We show that for any k ∈ Z + k \in {{\mathbf {Z}}^ + } \[ ∫ | f | 2 | lg ⁡ ( | f | ) | 2 s k / ( s + 1 ) d μ ⩽ c { ∑ | α | = 0 k ‖ D α f ‖ L 2 ( d μ ) 2 + ‖ f ‖ L 2 ( d μ ) 2 ∙ | lg ⁡ ( ‖ f ‖ L 2 ( d μ ) ) | 2 s k / ( s + 1 ) } \begin {array}{*{20}{c}} {\int {|f{|^2}|\lg (|f|){|^{2sk/(s + 1)}}d\mu } } \hfill \\ { \leqslant c\left \{ {\sum \limits _{|\alpha | = 0}^k {\left \|{D^\alpha }f\right \|_{{L_2}(d\mu )}^2 + \left \|f\right \|_{{L_2}(d\mu )}^2 \bullet |\lg (\left \|f\right \|_{{L_2}(d\mu )}){|^{2sk/(s + 1)}}} } \right \}} \hfill \\ \end {array} \] As a consequence we prove e − t ∇ ∗ ⋅ ∇ : L q ( R n , d μ ) → L p ( R n , d μ ) , p , q ≠ 1 , ∞ {e^{ - t{\nabla ^\ast } \cdot \nabla }}:{L_q}({{\mathbf {R}}^n},d\mu ) \to {L_p}({{\mathbf {R}}^n},d\mu ),p,q \ne 1,\infty , is bounded for all t > 0 t > 0 .

We give sufficient conditions for estimates of the form\[∫|u(x)|qdμ(x)⩽C‖u‖s,p1,u∈Hs,p,{\int {\left | {u(x)} \right |} ^q}d\mu (x) \leqslant C\left \| u \right \|_{s,p}^1,\qquad u \in {H^{s,p}},\]to hold, whereμ(x)\mu (x)is a measure and‖u‖s,p{\left \| u \right \|_{s,p}}is the norm of the Sobolev spaceHs,p{H^{s,p}}. Ifdμ=dxd\mu = dx, this reduces to the usual Sobolev inequality. The general form has much wider applications in both linear and nonlinear partial differential equations. An application is given in the last section.

We prove here that the Hardy space of B B -valued functions H 1 ( B ) {H^1}(B) defined by using the conjugate function and the one defined in terms of B B -valued atoms do not coincide for a general Banach space. The condition for them to coincide is the UMD property on B B . We also characterize the dual space of both spaces, the first one by using B ∗ {B^{\ast }} -valued distributions and the second one in terms of a new space of vector-valued measures, denoted B M O ( B ∗ ) \mathcal {B}\mathcal {M}\mathcal {O}({B^{\ast }}) , which coincides with the classical BMO ⁡ ( B ∗ ) \operatorname {BMO} ({B^{\ast }}) of functions when B ∗ {B^{\ast }} has the RNP.

A Banach space is flat if the girth of its unit ball is 4 and if the girth is achieved by some curve. (Equivalently, its unit ball can be circumnavigated along a centrally symmetric path whose length is 4.) Some basic geometric properties of flat Banach spaces are given. In particular, the term flat is justified.

We study the invariant subspace structure of the operator of multiplication by z z , M z {M_z} , on a class of Banach spaces of analytic functions. For operators on Hilbert spaces our class coincides with the adjoints of the operators in the Cowen-Douglas class B 1 ( Ω ¯ ) {\mathcal {B}_1}(\overline \Omega ) . We say that an invariant subspace M \mathcal {M} satisfies cod ⁡ M = 1 \operatorname {cod} \mathcal {M} = 1 if z M z\mathcal {M} has codimension one in M \mathcal {M} . We give various conditions on invariant subspaces which imply that cod ⁡ M = 1 \operatorname {cod} \mathcal {M} = 1 . In particular, we give a necessary and sufficient condition on two invariant subspaces M \mathcal {M} , N \mathcal {N} with cod ⁡ M = cod ⁡ N = 1 \operatorname {cod} \mathcal {M} = \operatorname {cod} \mathcal {N} = 1 so that their span again satisfies cod ⁡ ( M ∨ N ) = 1 \operatorname {cod} (\mathcal {M} \vee \mathcal {N}) = 1 . This result will be used to show that any invariant subspace of the Bergman space L a p , p ⩾ 1 L_a^p,\,p \geqslant 1 , which is generated by functions in L a 2 p L_a^{2p} , must satisfy cod ⁡ M = 1 \operatorname {cod} \mathcal {M} = 1 . For an invariant subspace M \mathcal {M} we then consider the operator S = M z ∗ | M ⊥ S = M_z^{\ast }|{\mathcal {M}^ \bot } . Under some extra assumption on the domain of holomorphy we show that the spectrum of S S coincides with the approximate point spectrum iff cod ⁡ M = 1 \operatorname {cod} \mathcal {M} = 1 . Finally, in the last section we obtain a structure theorem for invariant subspaces with cod ⁡ M = 1 \operatorname {cod} \mathcal {M} = 1 . This theorem applies to Dirichlet-type spaces.

In this paper we study the following problem: given a complete locally bounded sequence space Y, construct a locally bounded space Z with a subspace X such that both X and Z / X Z/X are isomorphic to Y, and such that X is uncomplemented in Z. We give a method for constructing Z under quite general conditions on Y, and we investigate some of the properties of Z. In particular, when Y is l p ( 1 > p > ∞ ) {l_p}\,(1\, > \,p\, > \,\infty ) , we identify the dual space of Z, we study the structure of basic sequences in Z, and we study the endomorphisms of Z and the projections of Z on infinite-dimensional subspaces.

Pseudo-differential operators with coefficients in Sobolev spaces H r , q , 1 ⩽ q ⩽ ∞ {H^{r,q}},1 \leqslant q \leqslant \infty , and their adjoints are studied on Hardy-Sobolev spaces H s , p , 0 > p ⩽ ∞ {H^{s,p}},\;0 > p \leqslant \infty . A symbolic calculus for these operators is developed, and the microlocal properties are studied. Finally, the invariance under coordinate transformations is proved.

We study subspaces of Besov spaces B p s , q B_p^{s,q} which are invariant under pointwise multiplication by characters. The case s > 0 s > 0 is completely described, and for the case s ⩽ 0 s \leqslant 0 we extend known results.

Combining combinatorial methods from set theory with the functional structure of certain Banach spaces we get some results on the isomorphic structure of nonseparable Banach spaces. The conclusions of the paper, in conjunction with already known results, give complete answers to problems of the theory of Banach spaces. An interesting point here is that some questions of Banach spaces theory are independent of Z.F.C. So, for example, the answer to a conjecture of Pełczynski that states that the isomorphic embeddability of L 1 { − 1 , 1 } α {L^1}{\{ - 1,\,1\} ^\alpha } into X ∗ {X^{\ast }} implies, for any infinite cardinal α \alpha , the isomorphic embedding of l α 1 l_\alpha ^1 into X X , gets the following form: if α = ω \alpha = \omega , has been proved from Pełczynski; if α > ω + \alpha > {\omega ^ + } , the proof is given in this paper; if α = ω + \alpha = {\omega ^ + } , in Z .F .C . + C .H . {\text {Z}}{\text {.F}}{\text {.C}}{\text {.}} + {\text {C}}{\text {.H}}{\text {.}} , an example discovered by Haydon gives a negative answer; if α = ω + \alpha = {\omega ^ + } , in Z .F .C . + ⌝ C .H . + M .A . {\text {Z}}{\text {.F}}{\text {.C}}{\text {.}} + \urcorner {\text {C}}{\text {.H}}{\text {.}} + {\text {M}}{\text {.A}}{\text {.}} , is also proved in this paper.

Given two Banach spaces F | | F|| and X | | | | X||\,|| , write F fr X iff F{\text { fr }}X{\text { iff}} for each finite-dimensional subspace F ′ F’ of F F and each number ε > 0 \varepsilon > 0 , there is an isomorphism V V of F ′ F’ into X X such that | | x | − | | V x | | | ≤ ε ||x| - ||Vx||| \leq \varepsilon for each x x in the unit ball of F ′ F’ . Given a property P {\mathbf {P}} of Banach spaces, X X is called super- P iff F fr X {\mathbf {P}}{\text { iff }}F{\text { fr }}X implies F F is P {\mathbf {P}} . Ergodicity and stability were defined in our articles On B B -convex Banach spaces, Math. Systems Theory 7 (1974), 294-299, and C. R. Acad. Sci. Paris Ser. A 275 (1972), 993, where it is shown that super-ergodicity and super-stability are equivalent to super-reflexivity introduced by R. C. James [Canad. J. Math. 24 (1972), 896-904]. Q Q -ergodicity is defined, and it is proved that super- Q Q -ergodicity is another property equivalent with super-reflexivity. A new proof is given of the theorem that J J -spaces are reflexive [Schaffer-Sundaresan, Math. Ann. 184 (1970), 163-168]. It is shown that if a Banach space X X is B B -convex, then each bounded sequence in X X contains a subsequence ( y n ) ({y_n}) such that the Cesàro averages of ( − 1 ) i y i {( - 1)^i}{y_i} converge to zero.

We show that on a starshaped domain Ω \Omega in C n {\operatorname {C} ^n} (actually on a somewhat larger, biholomorphically invariant class) the L p {\mathcal {L}^p} -Sobolev spaces of analytic functions form an interpolation scale for both the real and complex methods, for each p , 0 > p ⩽ ∞ p,\;0 > p \leqslant \infty . The case p = ∞ p = \infty gives the Lipschitz scale; here the functor ( , ) [ θ ] {(,)^{[\theta ]}} has to be considered (rather than ( , ) [ θ ] {(,)_{[\theta ]}} ).

Let E E be a Banach space. One often wants to measure how far E E is from being a Hilbert space. In this paper we define the skewness s ( E ) s(E) of a Banach space E E , 0 ⩽ s ( E ) ⩽ 2 0 \leqslant s(E) \leqslant 2 , which describes the asymmetry of the norm. We show that s ( E ) = s ( E ∗ ) s(E) = s({E^{\ast }}) for all Banach spaces E E . Further, s ( E ) = 0 s(E) = 0 if and only if E E is a (real) Hilbert space and s ( E ) = 2 s(E) = 2 if and only if E E is quadrate, so s ( E ) > 2 s(E) > 2 implies E E is reflexive. We discuss the computation of s ( L p ) s({L^p}) and describe its asymptotic behavior near p = 1 , 2 p = 1,2 and ∞ \infty . Finally, we discuss a higher-dimensional generalization of skewness which gives a characterization of smooth Banach spaces.

Let X X and Y Y be (infinite-dimensional) Banach spaces and denote their nonstandard hulls with respect to an ℵ 1 \aleph _1 -saturated enlargement by X ^ \hat X and Y ^ \hat Y respectively. If B ( X , Y ) \mathcal {B}(X,Y) denotes the space of bounded linear transformations then a subset S S of elements of B ( X , Y ) \mathcal {B}(X,Y) extends naturally to a subset S ^ \hat S of B ( X ^ , Y ^ ) \mathcal {B}(\hat {X}, \hat {Y}) . This paper studies the behaviour of various kinds of transformations under this extension and introduces, in this context, the concepts of super weakly compact, super strictly singular and socially compact operators. It shows that B ( X , Y ) ^ ⫋ B ( X ^ , Y ^ ) \widehat {\mathcal {B}(X,Y)} \subsetneqq \mathcal {B}(\hat {X}, \hat {Y}) provided X X and Y Y are infinite dimensional and contrasts this with the inclusion K ( H ^ ) ⫋ K ( H ) ^ \mathcal {K}(\hat H) \subsetneqq \widehat {\mathcal {K}(H)} where K ( H ) \mathcal {K}(H) denotes the space of compact operators on a Hilbert space.