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A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. Associated to a given continuous frame we construct certain Banach spaces. Many classical function spaces can be identified as such spaces. We provide a general method to derive Banach frames and atomic decompositions for these Banach spaces by sampling the continuous frame. This is done by generalizing the coorbit space theory developed by Feichtinger and Groechenig. As an important tool the concept of localization of frames is extended to continuous frames. As a byproduct we give a partial answer to the question raised by Ali, Antoine and Gazeau whether any continuous frame admits a corresponding discrete realization generated by sampling.

We give an overview on the c-function of a non-compactly causal symmetric space G/H and explain its interplay with harmonic analysis and representation theory.

We study function spaces and extension results in relation with Dirichlet problems involving integrodifferential operators. For such problems, data are prescribed on the complement of a given domain in the Euclidean space. We introduce a function space that serves as a trace space for nonlocal Dirichlet problems and study related extension results.

The main result says that every surjective isometry between two ideal Banach function spaces satisfying certain conditions can be presented as a composition of a measurable transformation of a variable and multiplication by a function.

Let $X$ be a nilpotent space such that there exists $N\geq 1$ with $H^N(X,\mathbb Q) \ne 0$ and $H^n(X,\mathbb Q)=0$ if $n>N$. Let $Y$ be a m-connected space with $m\geq N+1$ and $H^*(Y,\mathbb Q)$ is finitely generated as algebra. We assume that the odd part of the rational Hurewicz homomorphism: $\pi_{odd}(X)\otimes \mathbb Q\to H_{odd}(X,\mathbb Q)$ is non-zero. We prove that if the space $\mathcal F(X,Y)$ of continuous maps from $X$ to $Y$ is rationally formal, then $Y$ has the rational homotopy type of a finite product of Eilenberg Mac Lane spaces. At the opposite, we exhibit an example of a rationally formal space $\mathcal F(S^2,Y)$ where $Y$ is not rationally equivalent to a product of Eilenberg Mac Lane spaces.

We study a cut-off function lemma in projective spaces. We believe that this is well-known. We provide the details of the computation for later uses.

In this paper, we present a characterization of support functionals and smooth points in $L_{0}^{\Phi}$, the Musielak-Orlicz space equipped with the Orlicz norm. As a result, criterion for the smoothness of $L_{0}^{\Phi}$ is also obtained. Some expressions involving the norms of functionals in $(L_{0}^{\Phi})^{*}$, the topological dual of $L_{0}^{\Phi}$, are proved for arbitrary Musielak-Orlicz functions.

Let $L$ be a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$ whose heat kernels have the Gaussian upper bound estimates. Assume that the growth function $\varphi:\,\mathbb{R}^n\times[0,\infty) \to[0,\infty)$ satisfies that $\varphi(x,\cdot)$ is an Orlicz function and $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ (the class of uniformly Muckenhoupt weights). Let $H_{\varphi,\,L}(\mathbb{R}^n)$ be the Musielak-Orlicz-Hardy space introduced via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors obtain several maximal function characterizations of the space $H_{\varphi,\,L}(\mathbb{R}^n)$, which, especially, answer an open question of L. Song and L. Yan under an additional mild assumption satisfied by Schr\"odinger operators on $\mathbb{R}^n$ with non-negative potentials belonging to the reverse H\"older class, and second-order divergence form elliptic operators on $\mathbb{R}^n$ with bounded measurable real coefficients.

In the paper it is shown that there exist a function g from L1[0,1] and a weight function 0 = 1 with respect to signs-subseries of its Fourier-Walsh series.

We investigate Gevrey asymptotics for solutions to nonlinear parameter depending Cauchy problems with $2\pi$-periodic coefficients, for initial data living in a space of quasiperiodic functions. By means of the Borel-Laplace summation procedure, we construct sectorial holomorphic solutions which are shown to share the same formal power series as asymptotic expansion in the perturbation parameter. We observe a small divisor phenomenon which emerges from the quasiperiodic nature of the solutions space and which is the origin of the Gevrey type divergence of this formal series. Our result rests on the classical Ramis-Sibuya theorem which asks to prove that the difference of any two neighboring constructed solutions satisfies some exponential decay. This is done by an asymptotic study of a Dirichlet-like series whose exponents are positive real numbers which accumulate to the origin.

Let $B$ be a Blaschke product with zeros $\{a_n\}$. If $B' \in A^p_{\alpha}$ for certain $p$ and $\alpha$, it is shown that $\sum_n (1 - |a_n|)^{\beta} < \infty$ for appropriate values of $\beta$. Also, if $\{a_n\}$ is uniformly discrete and if $B' \in H^p$ or $B' \in A^{1+p}$ for any $p \in (0,1)$, it is shown that $\sum_n (1 - |a_n|)^{1-p} < \infty$.

In this paper we introduce a polynomial frame on the unit sphere $\sph$ of $\mathbb{R}^d$, for which every distribution has a wavelet-type decomposition. More importantly, we prove that many function spaces on the sphere $\sph$, such as $L^p$, $H^p$ and Besov spaces, can be characterized in terms of the coefficients in the wavelet decompositions, as in the usual Euclidean case $\mathbb{R}^d$. We also study a related nonlinear $m$-term approximation problem on $\sph$. In particular, we prove both a Jackson--type inequality and a Bernstein--type inequality associated to wavelet decompositions, which extend the corresponding results obtained by R. A. DeVore, B. Jawerth and V. Popov (``Compression of wavelet decompositions'', {\it Amer. J. Math.} {\bf 114} (1992), no. 4, 737--785).

In this paper we introduce a polynomial frame on the unit sphere $\sph$ of $\mathbb{R}^d$, for which every distribution has a wavelet-type decomposition. More importantly, we prove that many function spaces on the sphere $\sph$, such as $L^p$, $H^p$ and Besov spaces, can be characterized in terms of the coefficients in the wavelet decompositions, as in the usual Euclidean case $\mathbb{R}^d$. We also study a related nonlinear $m$-term approximation problem on $\sph$. In particular, we prove both a Jackson--type inequality and a Bernstein--type inequality associated to wavelet decompositions, which extend the corresponding results obtained by R. A. DeVore, B. Jawerth and V. Popov (``Compression of wavelet decompositions'', {\it Amer. J. Math.} {\bf 114} (1992), no. 4, 737--785).

Let $L$ be a nonnegative, self-adjoint operator satisfying Gaussian estimates on $L^2(\RR^n)$. In this article we give an atomic decomposition for the Hardy spaces $ H^p_{L,max}(\R)$ in terms of the nontangential maximal functions associated with the heat semigroup of $L$, and this leads eventually to characterizations of Hardy spaces associated to $L$, via atomic decomposition or the nontangential maximal functions. The proofs are based on a modification of technique due to A. Calder\'on \cite{C}.

We develop a framework for studying variational problems in Banach spaces with respect to gradient relations, which encompasses many of the notions of generalized gradients that appear in the literature. We stress the fact that our approach is not dependent on function spaces and therefore applies equally well to functions on metric spaces as to operator algebras. In particular, we consider analogues of Dirichlet and obstacle problems, as well as first eigenvalue problems, and formulate conditions for the existence of solutions and their uniqueness. Moreover, we investigate to what extent a lattice structure may be introduced on (ordered) Banach spaces via a norm-minimizing variational problem. A multitude of examples is provided to illustrate the versatility of our approach.

We study Whitehead products in the rational homotopy groups of a general component of a function space. For the component of any based map f: X \to Y, in either the based or free function space, our main results express the Whitehead product directly in terms of the Quillen minimal model of f. These results follow from a purely algebraic development in the setting of chain complexes of derivations of differential graded Lie algebras, which is of interest in its own right. We apply the results to study the Whitehead length of function space components.

This is the final version, to appear in Commentarii Mathematici Helvetici.

We compute the Green's function for the Hodge Laplacian on the symmetric spaces M\times\Sigma, where M is a simply connected n-dimensional Riemannian or Lorentzian manifold of constant curvature and \Sigma is a simply connected Riemannian surface of constant curvature. Our approach is based on a generalization to the case of differential forms of the method of spherical means and on the use of Riesz distributions on manifolds. The radial part of the Green's function is governed by a fourth order analogue of the Heun equation.

In this paper, we study the existence of infinite dimensional closed linear subspaces of a rearrangement invariant space on [0,1] every nonzero element of which does not belong to any included rearrangement invariant space of the same class such that the inclusion operator is disjointly strictly singular. We consider Lorentz, Marcinkiewicz and Orlicz spaces. The answer is affirmative for Marcinkiewicz spaces and negative for Lorentz and Orlicz spaces. Also, the same problem is studied for Nakano spaces assuming different hypothesis.

Given a map $f: X\rightarrow Y$ of simply connected spaces of finite type such. The space of based loops at $f$ of the space of maps between $X$ and $Y$ is denoted by $\Omega_{f} Map(X,Y)$. For $n> 0$, we give a model categorical interpretation of the existence (in functorial way) of an injective map of $\mathbb{Q}$-vector spaces $\pi_{n} \Omega_{f}Map(X,Y_{\mathbb{Q}}) \rightarrow HH^{-n}(C^{\ast}(Y),C^{\ast}(X)_{f})$, where $HH^{\ast}$ is the (negative) Hochschild cohomology and $C^{\ast}(X)_{f}$ is the rational cochain complex associated to $X$ equipped with a structure of $C^{\ast}(Y)$-differential graded bimodule via the induced map of differential graded algebras $f^{\ast}: C^{\ast}(Y)\rightarrow C^{\ast}(X)$. Moreover, we identifiy the image in presice way by using the Hodge filtration on Hochschild cohomology. In particular, when $X=Y$, we describe the fundamental group of the identity component of the monoid of self equivalence of a (rationalization of) space $X$ i.e., $\pi_{1} Aut(X_{\mathbb{Q}})_{id}$.

In this paper one-weight inequalities with general weights for Riemann-Liouville transform and $ n-$ dimensional fractional integral operator in variable exponent Lebesgue spaces defined on $\mathbb{R}^{n}$ are investigated. In particular, we derive necessary and sufficient conditions governing one-weight inequalities for these operators on the cone of non-negative decreasing functions in $L^{p(x)}$ spaces.

The interpolation property of Ces{\`a}ro sequence and function spaces is investigated. It is shown that $Ces_p(I)$ is an interpolation space between $Ces_{p_0}(I)$ and $Ces_{p_1}(I)$ for $1 < p_0 < p_1 \leq \infty$ and $1/p = (1 - \theta)/p_0 + \theta /p_1$ with $0 < \theta < 1$, where $I = [0, \infty)$ or $[0, 1]$. The same result is true for Ces{\`a}ro sequence spaces. On the other hand, $Ces_p[0, 1]$ is not an interpolation space between $Ces_1[0, 1]$ and $Ces_{\infty}[0, 1]$.

Montgomery's pair correlation conjecture predicts the asymptotic behavior of the function $N(T,\beta)$ defined to be the number of pairs $\gamma$ and $\gamma'$ of ordinates of nontrivial zeros of the Riemann zeta-function satisfying $0 0$, using Montgomery's formula and some extremal functions of exponential type. These functions are optimal in the sense that they majorize and minorize the characteristic function of the interval $[-\beta, \beta]$ in a way to minimize the $L^1\big(\mathbb{R}, \big\{1 - \big(\frac{\sin \pi x}{\pi x}\big)^2 \big\}\,dx\big)$-error. We give a complete solution for this extremal problem using the framework of reproducing kernel Hilbert spaces of entire functions. This extends previous work by P. X. Gallagher in 1985, where the case $\beta \in \frac12 \mathbb{N}$ was considered using non-extremal majorants and minorants.

In \cite{ds_hfs}, a geometric procedure for constructing a Nevanlinna-Pick problem on $\D^n$ with a specified set of uniqueness was established. In this sequel we conjecture a necessary and a sufficient condition for a Nevanlinna-Pick problem on $\DT$ to have a unique solution. We use the results of \cite{ds_hfs} and Bezout's theorem to establish three special cases of this conjecture.

We consider a broad family of test function spaces and their dual (distribution) space. The family includes Gelfand-Shilov spaces, a family of test function spaces introduced by S. Pilipovic. We deduce different characterizations of such spaces, especially under the Bargmann transform and the Short-time Fourier transform. The family also include a test function space, whose dual space is mapped by the Bargmann transform bijectively to the set of entire functions.

In this paper, we study anisotropic Bessel potential and Besov spaces, where the anisotropy measures the extra amount of regularity in certain directions. Some basic properties of these spaces are established along with applications to elliptic boundary value problems.

Some fixed point results are given for a class of functional contractions acting on (reflexive) triangular symmetric spaces. Technical connections with the corresponding theories over (standard) metric and partial metric spaces are also being established.

For pointed compact metric spaces $(X,d)$, we address the biduality problem as to when the space of Lipschitz functions $\mathrm{Lip}_0(X,d)$ is isometrically isomorphic to the bidual of the space of little Lipschitz functions $\mathrm{lip}_0(X,d)$, and show that this is the case whenever the closed unit ball of $\mathrm{lip}_0(X,d)$ is dense in the closed unit ball of $\mathrm{Lip}_0(X,d)$ with respect to the topology of pointwise convergence. Then we apply our density criterion to prove in an alternate way the real version of a classical result which asserts that $\mathrm{Lip}_0(X,d^\alpha)$ is isometrically isomorphic to $\mathrm{lip}_0(X,d^\alpha)^{**}$ for any $\alpha$ in $(0,1)$.

A working manuscript.

We study tightness properties and selective versions of separability in bitopological function spaces endowed with set-open topologies.

In this paper we propose a general coorbit space theory suitable to define coorbits of quasi-Banach spaces using an abstract continuous frame, indexed by a locally compact Hausdorff space, and an associated generalized voice transform. The proposed theory realizes a further step in the development of a universal abstract theory towards various function spaces and their atomic decompositions which has been initiated by Feichtinger and Gr{\"o}chenig in the late 1980ies. We combine the recent approaches in Rauhut, Ullrich and Rauhut to identify, in particular, various inhomogeneous (quasi-Banach) spaces of Besov-Lizorkin-Triebel type. To prove the potential of our new theory we apply it to spaces with variable smoothness and integrability which have attracted significant interest in the last 10 years. From the abstract discretization machinery we obtain atomic decompositions as well as wavelet frame isomorphisms for these spaces.

In this paper we propose a general coorbit space theory suitable to define coorbits of quasi-Banach spaces using an abstract continuous frame, indexed by a locally compact Hausdorff space, and an associated generalized voice transform. The proposed theory realizes a further step in the development of a universal abstract theory towards various function spaces and their atomic decompositions which has been initiated by Feichtinger and Gr{\"o}chenig in the late 1980ies. We combine the recent approaches in Rauhut, Ullrich and Rauhut to identify, in particular, various inhomogeneous (quasi-Banach) spaces of Besov-Lizorkin-Triebel type. To prove the potential of our new theory we apply it to spaces with variable smoothness and integrability which have attracted significant interest in the last 10 years. From the abstract discretization machinery we obtain atomic decompositions as well as wavelet frame isomorphisms for these spaces.

It is known that a continuous linear operator T defined on a Banach function space X(μ) (over a finite measure space (Ω,Σ,μ)) and with values in a Banach space X can be extended to a sort of optimal domain. Indeed, under certain assumptions on the space X(μ) and the operator T this optimal domain coincides with L1(mT), the space of all functions integrable with respect to the vector measure mT associated with T, and the optimal extension of T turns out to be the integration operator ImT. In this book the idea is taken up and the corresponding theory is translated to a larger class of function spaces, namely to Fr\'echet function spaces X(μ) (this time over a σ-finite measure space (Ω,Σ,μ). It is shown that under similar assumptions on X(μ) and T as in the case of Banach function spaces the so-called ``optimal extension process'' also works for this altered situation. In a further step the newly gained results are applied to four well-known operators defined on the Fréchet function spaces Lp-([0,1]) resp. Lp-(G) (where G is a compact Abelian group) and Lploc .

It is known that a continuous linear operator T defined on a Banach function space X(μ) (over a finite measure space (Ω,Σ,μ)) and with values in a Banach space X can be extended to a sort of optimal domain. Indeed, under certain assumptions on the space X(μ) and the operator T this optimal domain coincides with L1(mT), the space of all functions integrable with respect to the vector measure mT associated with T, and the optimal extension of T turns out to be the integration operator ImT. In this book the idea is taken up and the corresponding theory is translated to a larger class of function spaces, namely to Fr\'echet function spaces X(μ) (this time over a σ-finite measure space (Ω,Σ,μ). It is shown that under similar assumptions on X(μ) and T as in the case of Banach function spaces the so-called ``optimal extension process'' also works for this altered situation. In a further step the newly gained results are applied to four well-known operators defined on the Fréchet function spaces Lp-([0,1]) resp. Lp-(G) (where G is a compact Abelian group) and Lploc .

It is known that a continuous linear operator T defined on a Banach function space X(μ) (over a finite measure space (Ω,Σ,μ)) and with values in a Banach space X can be extended to a sort of optimal domain. Indeed, under certain assumptions on the space X(μ) and the operator T this optimal domain coincides with L1(mT), the space of all functions integrable with respect to the vector measure mT associated with T, and the optimal extension of T turns out to be the integration operator ImT. In this book the idea is taken up and the corresponding theory is translated to a larger class of function spaces, namely to Fr\'echet function spaces X(μ) (this time over a σ-finite measure space (Ω,Σ,μ). It is shown that under similar assumptions on X(μ) and T as in the case of Banach function spaces the so-called ``optimal extension process'' also works for this altered situation. In a further step the newly gained results are applied to four well-known operators defined on the Fréchet function spaces Lp-([0,1]) resp. Lp-(G) (where G is a compact Abelian group) and Lploc .

It is known that a continuous linear operator T defined on a Banach function space X(μ) (over a finite measure space (Ω,Σ,μ)) and with values in a Banach space X can be extended to a sort of optimal domain. Indeed, under certain assumptions on the space X(μ) and the operator T this optimal domain coincides with L1(mT), the space of all functions integrable with respect to the vector measure mT associated with T, and the optimal extension of T turns out to be the integration operator ImT. In this book the idea is taken up and the corresponding theory is translated to a larger class of function spaces, namely to Fr\'echet function spaces X(μ) (this time over a σ-finite measure space (Ω,Σ,μ). It is shown that under similar assumptions on X(μ) and T as in the case of Banach function spaces the so-called ``optimal extension process'' also works for this altered situation. In a further step the newly gained results are applied to four well-known operators defined on the Fréchet function spaces Lp-([0,1]) resp. Lp-(G) (where G is a compact Abelian group) and Lploc .

We establish square function estimates for integral operators on uniformly rectifiable sets by proving a local $T(b)$ theorem and applying it to show that such estimates are stable under the so-called big pieces functor. More generally, we consider integral operators associated with Ahlfors-David regular sets of arbitrary codimension in ambient quasi-metric spaces. The local $T(b)$ theorem is then used to establish an inductive scheme in which square function estimates on so-called big pieces of an Ahlfors-David regular set are proved to be sufficient for square function estimates to hold on the entire set. Extrapolation results for $L^p$ and Hardy space versions of these estimates are also established. Moreover, we prove square function estimates for integral operators associated with variable coefficient kernels, including the Schwartz kernels of pseudodifferential operators acting between vector bundles on subdomains with uniformly rectifiable boundaries on manifolds.

Motivated by recent applications of weighted norm inequalities to maximal regularity of first and second order Cauchy problems, we study real interpolation spaces on the basis of general Banach function spaces and, in particular, weighted rearrangement invariant Banach function spaces. We show equivalence of the trace method and the $K$-method, identify real interpolation spaces between a Banach space and the domain of a sectorial operator, and reprove an extension of Dore's theorem on the boundedness of $H^\infty$-functional calculus to this general setting.

Let $L$ be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients and $(p_-(L),\, p_+(L))$ be the maximal interval of exponents $q\in[1,\,\infty]$ such that the semigroup $\{e^{-tL}\}_{t>0}$ is bounded on $L^q(\mathbb{R}^n)$. In this article, the authors establish the non-tangential maximal function characterizations of the associated Hardy spaces $H_L^p(\mathbb{R}^n)$ for all $p\in(0,\,p_+(L))$, which, when $p=1$, answers a question asked by Deng et al. in [J. Funct. Anal. 263 (2012), 604-674]. Moreover, the authors characterize $H_L^p(\mathbb{R}^n)$ via various versions of square functions and Lusin-area functions associated to the operator $L$.

The optimal density function assigns to each symplectic toric manifold $M$ a number $0 < d \leq 1$ obtained by considering the ratio between the maximum volume of $M$ which can be filled by symplectically embedded disjoint balls and the total symplectic volume of $M$. In the toric version of this problem, $M$ is toric and the balls need to be embedded respecting the toric action on $M$. The goal of this note is first to give a brief survey of the notion of toric symplectic manifold and the recent constructions of moduli space structure on them, and recall how to define a natural density function on this moduli space. Then we review previous works which explain how the study of the density function can be reduced to a problem in convex geometry, and use this correspondence to to give a simple description of the regions of continuity of the maximal density function when the dimension is $4$.

Function (linear) spaces on which an arbitrary function operates (i.e. the space is stable w.r.t. the pointwise unary operation defined by the function) were investigated, for continuous real or complex operations, by deLeeuw-Katznelson, Sternfeld and Weit. They showed that, with suitable assumptions, a real or complex function space on which a non-affine continuous function operates is like an algebra. In this note, however, the point of view is somewhat different: the scalars are any field K, and from the outset no topology or continuity are assumed. Rather, a function space operated by a non-additively-affine (multivariate) function induces a topology on the set on which it is defined and the family of functions which operate on it induces topologies on K^n. These are used to derive a density property and to investigate "homomorphisms" of such spaces w.r.t. the "operations".

We investigate graph-directed iterated function systems in mixed Euclidean and p-adic spaces. Hausdorff measure and Hausdorff dimension in such spaces are defined, and an upper bound for the Hausdorff dimension is obtained. The relation between the Haar measure and the Hausdorff measure is clarified. Finally, we discus an example in ${Bbb R}\times{\Bbb Q}\sb 2$ and calculate upper and lower bounds for its Hausdorff dimension.

We describe briefly a new approach to some problems related to Teichm\"uller spaces, invariant metrics, and extremal quasiconformal maps. This approach is based on the properties of plurisubharmonic functions, especially of the plurisubharmonic Green function. The main theorem gives an explicit representation of the Green function for Teichm\"uller spaces by the Kobayashi-Teichm\"uller metric of these spaces. This leads to various applications. In particular, this gives a new characterization of extremal quasiconformal maps.

The graph topology $\tau_{\Gamma}$ is the topology on the space $C(X)$ of all continuous functions defined on a Tychonoff space $X$ inherited from the Vietoris topology on $X\times \mathbb R$ after identifying continuous functions with their graphs. It is shown that all completeness properties between complete metrizability and hereditary Baireness coincide for the graph topology if and only if $X$ is countably compact; however, the graph topology is $\alpha$-favorable in the strong Choquet game, regardless of $X$. Analogous results are obtained for the fine topology on $C(X)$. Pseudocompleteness, along with properties related to 1st and 2nd countability of $(C(X),\tau_{\Gamma})$ are also investigated.

In this paper, we present a natural method to prove that the centered Hardy-Littlewood maximal function, $M$, on real or complex hyperbolic spaces, $\mathbb{H}^n = \R^+ \times \R^{n - 1}$ and $\mathbb{H}_c^n = \R^+ \times \mathrm{H}(2(n - 1), 1)$, satisfies the inequality of the type $\| M f \|_p \leq A_p \| f \|_p$ for all $1 < p \leq +\infty$ and $f \in L^p$, where $A_p > 0$ is a constant independent of $n$. This method can easily be adapted to the case of harmonic $AN$ groups and would also be valid for noncompact symmetric spaces.

Let $G$ be a locally compact group and also let $H$ be a compact subgroup of $G$. It is shown that, if $\mu$ is a relatively invariant measure on $G/H$ then there is a well-defined convolution on $L^1(G/H,\mu)$ such that the Banach space $L^1(G/H,\mu)$ becomes a Banach algebra. We also find a generalized definition of this convolution for other $L^p$-spaces. Finally, we show that various types of involutions can be considered on $G/H$.

We consider the Gelfand and Kolmogorov numbers of compact embeddings between weighted function spaces of Besov and Triebel-Lizorkin type with polynomial weights in the non-limiting case. Our main purpose here is to complement our previous results in \cite{ZF10} in the context of the quasi-Banach setting, $0 < p, q \le \infty$. In addition, sharp estimates for their approximation numbers in several cases left open in Skrzypczak (JAT, 2005) are provided.

The Gel'fand-Shilov spaces of type S are considered as topological algebras with respect to the Moyal star product and their corresponding algebras of multipliers are defined and investigated. In contrast to the well-studied case of Schwartz's space S, these multipliers are allowed to have nonpolynomial growth or infinite order singularities. The Moyal multiplication is thereby extended to certain classes of ultradistributions, hyperfunctions, and analytic functionals. The main theorem of the paper characterizes those elements of the dual of a given test function space that are the Moyal multipliers of this space. The smallest nontrivial Fourier-invariant space in the scale of S-type spaces is shown to play a special role, because its corresponding Moyal multiplier algebra contains the largest algebra of functions for which the power series defining their star products are absolutely convergent. Furthermore, it contains analogous algebras associated with cone-shaped regions, which can be used to formulate a causality condition in quantum field theory on noncommutative space-time.

Generalising an analysis of Corvino and Schoen, we study surjectivity properties of the constraint map in general relativity in a large class of weighted Sobolev spaces. As a corollary we prove several perturbation, gluing, and extension results: we show existence of non-trivial, singularity-free, vacuum space-times which are stationary in a neighborhood of $i^0$; for small perturbations of parity-covariant initial data sufficiently close to those for Minkowski space-time this leads to space-times with a smooth global Scri; we prove existence of initial data for many black holes which are exactly Kerr -- or exactly Schwarzschild -- both near infinity and near each of the connected components of the apparent horizon; under appropriate conditions we obtain existence of vacuum extensions of vacuum initial data across compact boundaries; we show that for generic metrics the deformations in the Isenberg-Mazzeo-Pollack gluings can be localised, so that the initial data on the connected sum manifold coincide with the original ones except for a small neighborhood of the gluing region; we prove existence of asymptotically flat solutions which are static or stationary up to $r^{-m}$ terms, for any fixed $m$, and with multipole moments freely prescribable within certain ranges.

Let $s_{n}(T)$ denote the $n$th approximation, isomorphism, Gelfand, Kolmogorov or Bernstein number of the Hardy-type integral operator $T$ given by $$ Tf(x)=v(x)\int_{a}^{x}u(t)f(t)dt,\,\,\,x\in(a,b)\,\,(-\infty 0$ depend only on the space $E.$