Aggrandization of spaces: a new general approach and application

We show that grand spaces of holomorphic functions may be equivalently defined in terms of aggrandization related only to the boundary. We base ourselves on recent studies of the so-called local aggrandization of Lebesgue spaces and extent this approach to the case of arbitrary Banach spaces of functions on metric spaces. We apply this approach to prove, in the case of Bergman and Bergman–Morrey spaces on the unit disk, that these grand spaces may be equivalently defined as grand spaces with weighted aggrandization on the boundary.

We consider holomorphic functions $\phi$ taking the unit disc $U$ into itself, and Banach spaces $X$ consisting of functions holomorphic in $U$ and continuous on its closure; and show that under some natural hypotheses on $X$: if $\phi$ induces a compact composition operator on $X$, then $\phi (U)$ must be a relatively compact subset of $U$. Spaces $X$ which satisfy the hypotheses of this theorem include the disc algebra, "heavily" weighted Dirichlet spaces, spaces of holomorphic Lipschitz functions, and the space of functions with derivative in a Hardy space ${H^p}(p \geq 1)$. It is well known that the theorem is not true for "large" spaces such as the Hardy and Bergman spaces. Surprisingly, it also fails in "very small spaces," such as the Hilbert space of holomorphic functions $f(z) = \sum {{a_n}{z^n}}$ determined by the condition $\sum {|{a_n}{|^2}\exp \left ( {\sqrt n } \right ) < \infty }$. The property of Möbiusinvariance plays a crucial and mysterious role in these matters.

In recent years, it had become apparent that the plethora of existing function spaces were not sufficient to model a wide variety of applications, e.g., in the modeling of electrorheological fluids, thermorheological fluids, in the study of image processing, in differential equations with nonstandard growth, among others. Thus, naturally, new fine scales of function spaces have been introduced, namely variable exponent spaces and grand spaces. In this chapter we study variable exponent Lebesgue spaces and grand Lebesgue spaces. In variable exponent Lebesgue spaces we study the problem of normability, denseness, completeness, embedding, among others. We give a brisk introduction to grand Lebesgue spaces via Banach function space theory, dealing with the problem of normability, embeddings, denseness, reflexivity, and the validity of a Hardy inequality in the aforementioned spaces.

This survey is aimed at the audience of readers interested in the information on mapping properties of various forms of fractional integration operators, including multidimensional ones, in a large scale of various known function spaces. As is well known, the fractional integrals defined in this or other forms improve in some sense the properties of the functions, at least locally, while fractional derivatives to the contrary worsen them. With the development of functional analysis this simple fact led to a number of important results on the mapping properties of fractional integrals in various function spaces. In the one-dimensional case we consider both Riemann-Liouville and Liouville forms of fractional integrals and derivatives. In the multidimensional case we consider in particular mixed Liouville fractional integrals, Riesz fractional integrals of elliptic and hyperbolic type and hypersingular integrals. Among the function spaces considered in this survey, the reader can find Hölder spaces, Lebesgue spaces, Morrey spaces, Grand spaces and also weighted and/or variable exponent versions.

The purpose of this special issue was to present new developments in the theory of function spaces, along with the deep interconnections with approximation theory and the applications in various fields of pure and applied mathematics. The reaction of the mathematical community was very satisfactory. We collected thirty-five submissions, covering a wide range of mathematical topics, ten of which were found to be suitable for publications in this issue. The major part of the accepted papers treats function spaces and their applications. In this respect, in the article by X Yang et al. a new class of function spaces, named “multi-βnormed spaces”, is introduced, in connection with stability properties of certain type of functional equations, while, in the paper by A. A. Bakery, sequential spaces of Orlicz type are studied and connected with the theory of summability. In the review paper by L. Angeloni and G. Vinti, the approximation theory in the space of functions with bounded variation is developed, in view of applications to signal processing. Different notions of variation are considered and several approximation theorems for families of integral or discrete type operators are given. In the more theoretical article by S. Wulede et al., a new class of Banach spaces which generalizes the class of uniformly extremely convex Banach spaces is introduced, and some characterizations of these spaces are given. Another paper by N. Khan treats the convergence of new type of double sequences, here introduced, in n-normed spaces. An interesting abstract approach to the theory of filter convergence is given in the article by A. Boccuto and X. Dimitriou, in which the links with function spaces and approximation theory are also dealt with. Other aspects of the theory of function spaces and their interconnections with calculus of variations, numerical analysis, complex variables, and stochastic processes are discussed, respectively, in the articles by T. Ma and Y. Feng, H. Wang et al., S. Wang and T. Zhan, and finally P. Duan.These four papers point out how certain methods of general approximation theory in function spaces can be employed in order to solve problems coming from a large variety of mathematical fields. We think that these contributions may represent starting points for new researches in the field of function spaces and approximation theory.

Given a Banach space X, we consider Hardy spaces of X-valued functions on the infinite polytorus, Hardy spaces of X-valued Dirichlet series (defined as the image of the previous ones by the Bohr transform), and Hardy spaces of X-valued holomorphic functions on l_2 ∩ B_{c0}. The chapter is dedicated to study the interplay between these spaces. It is shown that the space of functions on the polytorus always forms a subspace of the one of holomorphic functions, and these two are isometrically isomorphic if and only if X has ARNP. Then the question arises of what do we find in the side of Dirichlet series when we look at the image of the Hardy space of holomorphic functions. This is also answered, showing that this consists of Dirichlet series for which all horizontal translations (those whose coefficients are (a_n/n^ε)) are in \mathcal{H}_p with uniformly bounded norms. Also, a version of the brothers Riesz theorem for vector-valued functions is given.

Duality Theory for Spaces of Germs and Holomorphic Functions on Nuclear Spaces

The so called grand spaces nowadays are one of the main objects in the theory of function spaces. Grand Lebesgue spaces were introduced by T. Iwaniec and C. Sbordone in the case of sets $\Omega$ with finite measure $|\Omega|&lt;\infty$, and by the authors in the case $|\Omega|=\infty$. The latter is based on introduction of the notion of grandizer. The idea of "grandization" was also applied in the context of Morrey spaces. In this paper we develop the idea of grandization to more general Morrey spaces $L^{p,q,w}(\mathbb{R}^n)$, known as Morrey type spaces. We introduce grand Morrey type spaces, which include mixed and partial grand versions of such spaces. The mixed grand space is defined by the norm $$ \sup_{\varepsilon,\delta} \varphi(\varepsilon,\delta)\sup_{x\in E} \left(\int\limits_{0}^{\infty}{w(r)^{q-\delta}}b(r)^{\frac{\delta}{q}} \left(\,\int\limits_{|x-y|&lt;r}\big|f(y)\big|^{p-\varepsilon} a(y)^{\frac{\varepsilon}{p}}\,dy\right)^{\frac{q-\delta}{p-\varepsilon}} \frac{dr}{r}\right)^{\frac{1}{q-\varepsilon}} $$ with the use of two grandizers $a$ and $b$. In the case of grand spaces, partial with respect to the exponent $q$, we study the boundedness of some integral operators. The class of these operators contains, in particular, multidimensional versions of Hardy type and Hilbert operators.

This workshop brought together researchers studying a variety of problems related to the homotopy theory of function spaces. Topics covered included: evaluation maps and Gottlieb groups, the classification of gauge groups and of other function space components, algebraic models for function spaces both in the rational and in the p-local settings, operads, configuration spaces, free and based loop spaces and infinite-dimensional Lie groups. Mathematics Subject Classification (2000): 55P48, 55P60, 55P62, 22E65. Introduction by the Organisers The study of function spaces from an algebraic topological point of view dates back, at least, to the 1950s. G. Whitehead posed the basic problem of classifying the path components of a function space up to homotopy type and obtained the first results on this problem as an early application of the Whitehead product. Subsequent work of Thom and Federer paved the way for the computation of algebraic invariants of function spaces. In the late 1960s, Gottlieb initiated the study of the evaluation map, the evaluation subgroups and, in particular, the Gottlieb groups of space. In the 1970s, Hansen, Moller, Sutherland and others studied the homotopy classification problem for the components of a function space with many complete results. An early, famous application of Sullivan’s rational homotopy theory, the Vigue-Sullivan model for the free loop space of a manifold, led to a solution of the closed geodesic problem and showed the power of Sullivan’s algebraic models for homotopy theory. The 1980s saw steady progress on function spaces, especially in the local settings. Following Sullivan’s sketch, Haefliger described a model for the rational homotopy type of the space of sections of a nilpotent fibration. Felix, Halperin 2 Oberwolfach Report 19/2009 and Thomas obtained global results on the vanishing and dimension of rationalized Gottlieb groups. Brown, Peterson and L. Smith gave a second rational model for function spaces in terms of Lannes’s division functor. Finally and notably, Miller published his celebrated proof of the Sullivan conjecture concerning the contractibility of certain functions spaces during this period, a major advance in homotopy theory. In recent years, the study of function spaces and related topics has expanded and accelerated. Whitehead’s original classification problem is actively researched in the context of gauge groups. Gottlieb groups remain a challenging computation problem in the integral setting and, after rationalization, are the subject of a basic conjecture in rational homotopy theory. The study of the free loop space of a manifold has undergone a renaissance with the discovery of Chas-Sullivan string topology. Meanwhile, the further development of algebraic models for the rational and p-local homotopy theory of function spaces has opened the field to whole new types of questions, computations and, significantly, applications of function space techniques in other areas of homotopy theory. This workshop included 23 mathematicians with expertise and active research programs in these various areas. In addition to specialized talks, there were several invited survey talks on broad topics including Gottlieb groups, gauge groups and algebraic models for function spaces after localization. There were two extended problem sessions. Homotopy Theory of Function Spaces and Related Topics 3 Workshop: Homotopy Theory of Function Spaces and Related Topics

This volume on “Topics in Function Spaces, Differential Operators, Harmonic and Fractal Analysis” is dedicated to our teacher and friend Hans Triebel on the occasion of his seventy-fifth birthday. Hans Triebel was born on February 7, 1936 in Dessau, Germany. He studied mathematics and physics at the Friedrich-Schiller-University of Jena from 1954 to 1959, graduating with a Diploma Degree in mathematics. At the beginning of his academic career he worked in classical complex analysis and received a Ph.D. in Mathematics at the Friedrich-Schiller-University in 1962. Motivated by an interest in both mathematics and physics, he studied Sobolev's famous 1950 book and learned about the theory of distributions as developed by L. Schwartz. This might have been the catalyst for his change of research topic towards partial differential operators and function spaces. As a postdoc he spent one year at the University of Leningrad (St. Petersburg), where he enjoyed the intellectual ferment of the atmosphere created by such great mathematicians as Uraltseva, Ladyzhenskaya, Birman and Solomyak, and attended lectures by Birman on functional analysis, spectral theory and quantum mechanics. Inspired by the Russian School of Mathematics he focused his research on recent developments in linear and nonlinear partial differential operators, spectral theory and functional analysis, rapidly obtaining far-reaching results with a deep impact on further research in this field. In particular, he realized the significance of function spaces and contributed to both theory and applications in a decisive way. He completed his Habilitation Thesis on function spaces and nonlinear analysis in 1966, becoming Full Professor of Analysis at the Friedrich-Schiller-University in 1970. His further studies were also strongly influenced and motivated by personal contacts with S. G. Krein, J. Peetre, as well as by new approaches to the theory of function spaces based on Fourier-analytical techniques due to S. M. Nikol'skij, E. M. Stein and C. Fefferman. The development of the modern theory of function spaces in the last 40 years and its application to various branches in both pure and applied mathematics owes much to his seminal contributions. The bare facts are impressive: he has published more than 200 papers in internationally acknowledged journals, and has written no less than 18 monographs and textbooks; the rate of production of new and interesting results shows no sign of decreasing! Perhaps he is best known by the series of books he has written which present systematic treatments of the theory of function spaces from different points of view, thus revealing its symbiotic relationship with interpolation theory, harmonic analysis, partial differential equations, nonlinear operators, entropy, spectral theory, fractal analysis, wavelet theory, and theoretical numerical analysis. In particular, his books Interpolation Theory, Differential Operators, Function Spaces (finished in 1974 and published in 1978) as well as Theory of Function Spaces (based on earlier lecture notes and published in 1983) are much-quoted standard references and have been translated into Russian. At the textbook level, his Higher Analysis and Analysis and Mathematical Physics are masterpieces, relating mathematical theory to physical applications in an extraordinarily convincing way, and made even more vivid in a series of remarkable lecture courses at Jena. Hans Triebel has supervised nearly 40 Ph.D. students, many of whom have become internationally recognised mathematicians. He is on the editorial boards of various international journals, and in particular was an editor of Mathematische Nachrichten for many years. The reputation of the analysis department at the university of Jena owes much to his pioneering scientific work and the activities of his research group on function spaces. His outstanding scientific achievements were recognised by a National Award of the German Democratic Republic for Science and Technology in 1983 and the award of a Doctor of Science honoris causa by the University of Sussex in 1990. He was elected as (Corresponding) Member of the Academy of Science of the German Democratic Republic in 1978. Since 1993 he has been a Member of the Berlin-Brandenburg Academy of Science (formerly the Prussian Academy of Science). No less remarkable than his mathematical ability are his personal qualities, coupling total integrity, resolution and great warmth with an irreverent sense of humour; stories abound of his preparedness to lecture very early in the morning, sometimes to the surprise of the students! We are glad to have this opportunity to express our deep gratitude to him for sharing with so many of his colleagues his ideas and encyclopaedic knowledge. The present collection of papers is a tribute to his distinguished work and reflects recent developments in the theory of function spaces and related fields by outstanding experts. It is a pleasure to thank all the authors for their contributions.

We consider holomorphic functionsϕ\phitaking the unit discUUinto itself, and Banach spacesXXconsisting of functions holomorphic inUUand continuous on its closure; and show that under some natural hypotheses onXX:ifϕ\phiinduces a compact composition operator onXX,thenϕ(U)\phi (U)must be a relatively compact subset ofUU. SpacesXXwhich satisfy the hypotheses of this theorem include the disc algebra, "heavily" weighted Dirichlet spaces, spaces of holomorphic Lipschitz functions, and the space of functions with derivative in a Hardy spaceHp(p≥1){H^p}(p \geq 1). It is well known that the theorem isnottrue for "large" spaces such as the Hardy and Bergman spaces. Surprisingly, it also fails in "very small spaces," such as the Hilbert space of holomorphic functionsf(z)=∑anznf(z) = \sum {{a_n}{z^n}}determined by the condition∑|an|2exp⁡(n)&gt;∞\sum {|{a_n}{|^2}\exp \left ( {\sqrt n } \right ) &gt; \infty }. The property of Möbiusinvariance plays a crucial and mysterious role in these matters.

The Ph.D. Thesis ¿Operators on weighted spaces of holomorphic functions¿ presented&#13;\nhere treats different areas of functional analysis such as spaces of holomorphic&#13;\nfunctions, infinite dimensional holomorphy and dynamics of operators.&#13;\nAfter a first chapter that introduces the notation, definitions and the basic results&#13;\nwe will use throughout the thesis, the text is divided into two parts. A first one,&#13;\nconsisting of Chapters 1 and 2, focused on a study of weighted (LB)-spaces of entire&#13;\nfunctions on Banach spaces, and a second one, corresponding to Chapters 3 and&#13;\n4, where we consider differentiation and integration operators acting on different&#13;\nclasses of weighted spaces of entire functions to study its dynamical behaviour. In&#13;\nwhat follows, we give a brief description of the different chapters:&#13;\nIn Chapter 1, given a decreasing sequence of continuous radial weights on a Banach&#13;\nspace X, we consider the weighted inductive limits of spaces of entire functions&#13;\nVH(X) and VH0(X). Weighted spaces of holomorphic functions appear naturally&#13;\nin the study of growth conditions of holomorphic functions and have been investigated&#13;\nby many authors since the work of Williams in 1967, Rubel and Shields&#13;\nin 1970 and Shields and Williams in 1971. We determine conditions on the family&#13;\nof weights to ensure that the corresponding weighted space is an algebra or&#13;\nhas polynomial Schauder decompositions. We study Hörmander algebras of entire&#13;\nfunctions defined on a Banach space and we give a description of them in terms of&#13;\nsequence spaces. We also focus on algebra homomorphisms between these spaces&#13;\nand obtain a Banach-Stone type theorem for a particular decreasing family of&#13;\nweights. Finally, we study the spectra of these weighted algebras, endowing them&#13;\nwith an analytic structure, and we prove that each function f ¿ VH(X) extends&#13;\nnaturally to an analytic function defined on the spectrum. Given an algebra homomorphism,&#13;\nwe also investigate how the mapping induced between the spectra&#13;\nacts on the corresponding analytic structures and we show how in this setting&#13;\ncomposition operators have a different behavior from that for holomorphic functions&#13;\nof bounded type. This research is related to recent work by Carando, García,&#13;\nMaestre and Sevilla-Peris. The results included in this chapter are published by&#13;\nBeltrán in [14]. Chapter 2 is devoted to study the predual of VH(X) in order to linearize this space&#13;\nof entire functions. We apply Mujica¿s completeness theorem for (LB)-spaces to&#13;\nfind a predual and to prove that VH(X) is regular and complete. We also study&#13;\nconditions to ensure that the equality VH0(X) = VH(X) holds. At this point,&#13;\nwe will see some differences between the finite and the infinite dimensional cases.&#13;\nFinally, we give conditions which ensure that a function f defined in a subset&#13;\nA of X, with values in another Banach space E, and admitting certain weak&#13;\nextensions in a space of holomorphic functions can be holomorphically extended&#13;\nin the corresponding space of vector-valued functions. Most of the results obtained&#13;\nhave been published by the author in [13].&#13;\nThe rest of the thesis is devoted to study the dynamical behaviour of the following&#13;\nthree operators on weighted spaces of entire functions: the differentiation operator&#13;\nDf(z) = f (z), the integration operator Jf(z) = z&#13;\n0 f(¿)d¿ and the Hardy&#13;\noperator Hf(z) = 1&#13;\nz z&#13;\n0 f(¿)d¿, z ¿ C.&#13;\nIn Chapter 3 we focus on the dynamics of these operators on a wide class of&#13;\nweighted Banach spaces of entire functions defined by means of integrals and&#13;\nsupremum norms: the weighted spaces of entire functions Bp,q(v), 1 ¿ p ¿ ¿,&#13;\nand 1 ¿ q ¿ ¿. For q = ¿ they are known as generalized weighted Bergman&#13;\nspaces of entire functions, denoted by Hv(C) and H0&#13;\nv (C) if, in addition, p = ¿.&#13;\nWe analyze when they are hypercyclic, chaotic, power bounded, mean ergodic&#13;\nor uniformly mean ergodic; thus complementing also work by Bonet and Ricker&#13;\nabout mean ergodic multiplication operators. Moreover, for weights satisfying&#13;\nsome conditions, we estimate the norm of the operators and study their spectrum.&#13;\nSpecial emphasis is made on exponential weights. The content of this chapter is&#13;\npublished in [17] and [15].&#13;\nFor differential operators ¿(D) : Bp,q(v) ¿ Bp,q(v), whenever D : Bp,q(v) ¿&#13;\nBp,q(v) is continuous and ¿ is an entire function, we study hypercyclicity and&#13;\nchaos. The chapter ends with an example provided by A. Peris of a hypercyclic&#13;\nand uniformly mean ergodic operator. To our knowledge, this is the first example&#13;\nof an operator with these two properties. We thank him for giving us permission&#13;\nto include it in our thesis.&#13;\nThe last chapter is devoted to the study of the dynamics of the differentiation and&#13;\nthe integration operators on weighted inductive and projective limits of spaces of&#13;\nentire functions. We give sufficient conditions so that D and J are continuous on&#13;\nthese spaces and we characterize when the differentiation operator is hypercyclic,&#13;\ntopologically mixing or chaotic on projective limits. Finally, the dynamics of these&#13;\noperators is investigated in the Hörmander algebras Ap(C) and A0&#13;\np(C). The results&#13;\nconcerning this topic are included by Bonet, Fernández and the author in [16].

Let D be a Cartan domain in Cd and let G = Aut(D) be the group of all biholomorphic automorphisms of G. Consider the projective representation of G on spaces of holomorphic functions on D ((Uv(g)f)(z) {j(g-1))((z))) {j(g -1)(z)}(z)(1())g∈G where z is the genus of D and W is in the Wallach set D. We identify the minimal and the maximal Uv ((G))-invariant Banach spaces of holomorphic functions on D in a very explicit way: The minimal space 𝔐v is a Besov-1 space, and the maximal space Mv is a weighted 8-space. Moreover, with respect to the pairing under the (unique) U(v)(Uv)- invariant inner product we have 𝔐v G =Mv. In the second part of the paper we consider invariant Banach spaces of vector-valued holomorphic functions and obtain analogous descriptions of the unique maximal and minimal space, in particular for the important special case of “constant” partitions which arises naturally in connection with nontube type domains.

Compactness is undoubtedly one of the central and most relevant notions in mathematics. The present mini-workshop was centered around some important compactness problems connected to interpolation theory and to the theory of function spaces, two very closely related areas. Our aim was to bring together leading experts and active younger researchers in these fields. The mini-workshop was attended by 16 participants from Germany (5), Spain (5), Israel (2), Poland (2), Sweden (1) and United States (1), and the main problems we dealt with can, more specifically, be grouped as follows. 1.1. Complex interpolation of compact operators 1.2. Real interpolation of compactness and similar properties 2.1. Entropy numbers of such embeddings and applications to spectral theory of differential operators 2.2. Entropy techniques in sequence spaces Let us shortly describe these topics. 1.1. An outstanding problem in interpolation theory is the question, whether the complex interpolation method preserves compactness of operators. This is open since 40 years, and by now only partial answers are known. Recently a new and promising general approach has been proposed. We discussed this approach. 1.2. For the real interpolation method, however, it is well-known that it does preserve compactness. So it is quite natural to ask for quantitative versions of this purely qualitative result, for instance in terms of the measure of non-compactness, or in terms of entropy numbers. Another natural question is to study whether similar properties, like weak compactness, for example, are stable under real interpolation as well. These problems were addressed in some talks. 2.1. The sequence of entropy numbers (e_k(T))_{k=1}^\infty of a bounded linear operator T between quasi-Banach spaces can be considered as a quantification of compactness, since T is compact if and only if \lim_{k\to\infty} e_k(T)=0 . The basis for applications to spectral theory is the famous Carl–Triebel inequality, which relates entropy numbers of Riesz operators to its eigenvalues. Many concrete problems lead to the investigation of compact embeddings of certain function spaces, e.g. Sobolev or Besov spaces. In the talks both a survey on the general framework as well as new entropy estimates for specific embeddings were given. 2.2. Using various methods, for instance wavelet or atomic decompositions, the function space embeddings can very often be reduced to embeddings of (fairly complicated) sequence spaces. For the estimation of their entropy numbers one needs many different techniques, some of them quite new. Such techniques also were the subject of talks. Finally, several other aspects of interpolation were treated in talks, e.g. approximation spaces, bilinear interpolation, relation to eigenvalues and operator ideals. We list the abstracts of all talks in chronological order. The scientific program started with two survey lectures by Triebel and Cwikel, leading experts in their fields, followed by two more survey-style talks of the organisers. Then all other participants reported on own recent research results. In addition to this “official” program, which was already scheduled in advance, there was a number of further activities. Several participants offered a second talk, on another topic of common interest, or continued their respective talks in order to explain some technicalities in greater detail. The remaining time was used for many intensive discussions in smaller groups, and on Friday a problem session was held. The aim was to summarize the results of the mini-workshop and to discuss and collect several relevant problems, thus pointing out possible directions for further research in our field. Concerning social activities, one should mention the traditional hiking tour to St. Roman on Wednesday afternoon and the, maybe less traditional, joint session of all three parallel mini-workshops. The aim of this informal interdisciplinary session was to explain very briefly the kind of problems and ideas of our respective areas. Before this meeting there were serious doubts whether the intended goal would be achievable in only a few minutes, but afterwards it was general opinion that we have had a surprisingly inspiring and interesting evening, giving in fact a rough impression of the other two research areas. Last but not least, the organisers would like to express their gratitude to the director and the authorities of the Mathematisches Forschungsinstitut Oberwolfach for making this mini-workshop possible and for the constant support in its organisation and preparation. On behalf of all participants we thank all members of the staff for creating the unique working atmosphere, which made our stay so pleasant and which contributed substantially to the success of our mini-workshop.

Weighted spaces of harmonic and holomorphic functions on the unit disc are studied. We show that for all radial weights which are not decreasing too fast the space of harmonic functions is isomorphic to c0. For the weights that we consider we completely characterize those spaces of holomorphic functions which are isomorphic to c0. Moreover, we determine when the Riesz projection, mapping the weighted space of harmonic functions onto the corresponding space of holomorphic functions, is bounded.