We use a special tiling for the hyperbolic d-space $\mathbb {H}^d$ for $d=2,3,4$ to construct an (almost) explicit isomorphism between the Lipschitz-free space $\mathcal {F}(\mathbb {H}^d)$ and $\mathcal {F}(P)\oplus \mathcal {F}(\mathcal {N})$, where P is a polytope in $\mathbb {R}^d$ and $\mathcal {N}$ a net in $\mathbb {H}^d$ coming from the tiling. This implies that the spaces $\mathcal {F}(\mathbb {H}^d)$ and $\mathcal {F}(\mathbb {R}^d)\oplus \mathcal {F}(\mathcal {M})$ are isomorphic for every net $\mathcal {M}$ in $\mathbb {H}^d$. In particular, we obtain that, for $d=2,3,4$, $\mathcal {F}(\mathbb {H}^d)$ has a Schauder basis. Moreover, using a similar method, we also give an explicit isomorphism between $\mathrm {Lip}(\mathbb {H}^d)$ and $\mathrm {Lip}(\mathbb {R}^d)$.

Let $X, Y$ be two locally compact Hausdorff spaces and $T:C_0(X)\rightarrow C_0(Y)$ be a standard surjective ɛ-norm-additive map, i.e.

\begin{equation*}\big|\|T(f)+T(g)\|-\|f+g\|\big|\leq \varepsilon,\;{\rm for\;all}\; f, g\in C_0(X).\end{equation*}

Then there exist a homeomorphism $\varphi:Y\rightarrow X$ and a continuous function $\lambda:Y\rightarrow\lbrace\pm1\rbrace$ such that

\begin{equation*}|T(f)(y)-\lambda(y)f(\varphi(y))|\leq\frac{3}{2}\varepsilon,\;{\rm for\;all}\;y\in Y,\;f\in C_0(X).\end{equation*}

The estimate ‘$\frac{3}{2}\varepsilon$’ is optimal. And this result can be regarded as a new nonlinear extension of the Banach–Stone theorem.

Let $B(\Omega )$ be a Banach space of holomorphic functions on a bounded connected domain $\Omega $ in ${{\mathbb C}^n}$. In this paper, we establish a criterion for $B(\Omega )$ to be reflexive via evaluation functions on $B(\Omega )$, that is, $B(\Omega )$ is reflexive if and only if the evaluation functions span the dual space $(B(\Omega ))^{*} $.

Let X be a compact metric space, C(X) be the space of continuous real-valued functions on X and $A_{1},A_{2}$ be two closed subalgebras of C(X) containing constant functions. We consider the problem of approximation of a function $f\in C(X)$ by elements from $A_{1}+A_{2}$. We prove a Chebyshev-type alternation theorem for a function $u_{0} \in A_{1}+A_{2}$ to be a best approximation to f.

We construct a nonseparable Banach space $\mathcal {X}$ (actually, of density continuum) such that any uncountable subset $\mathcal {Y}$ of the unit sphere of $\mathcal {X}$ contains uncountably many points distant by less than $1$ (in fact, by less then $1-\varepsilon $ for some $\varepsilon>0$). This solves in the negative the central problem of the search for a nonseparable version of Kottman’s theorem which so far has produced many deep positive results for special classes of Banach spaces and has related the global properties of the spaces to the distances between points of uncountable subsets of the unit sphere. The property of our space is strong enough to imply that it contains neither an uncountable Auerbach system nor an uncountable equilateral set. The space is a strictly convex renorming of the Johnson–Lindenstrauss space induced by an $\mathbb {R}$-embeddable almost disjoint family of subsets of $\mathbb {N}$. We also show that this special feature of the almost disjoint family is essential to obtain the above properties.

Given a holomorphic self-map $\varphi $ of $\mathbb {D}$ (the open unit disc in $\mathbb {C}$ ), the composition operator $C_{\varphi } f = f \circ \varphi $ , $f \in H^2(\mathbb {\mathbb {D}})$ , defines a bounded linear operator on the Hardy space $H^2(\mathbb {\mathbb {D}})$ . The model spaces are the backward shift-invariant closed subspaces of $H^2(\mathbb {\mathbb {D}})$ , which are canonically associated with inner functions. In this paper, we study model spaces that are invariant under composition operators. Emphasis is put on finite-dimensional model spaces, affine transformations, and linear fractional transformations.

We construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalising and sharpening estimates and adapting the calculus to the angle of sectoriality. The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. To achieve this, we develop the theory of associated function spaces in ways that are interesting and significant. As consequences of our calculi, we derive several well-known operator norm estimates and provide generalisations of some of them.

Let $\mathrm {Lip}_0(M)$ be the space of Lipschitz functions on a complete metric space M that vanish at a base point. We prove that every normal functional in ${\mathrm {Lip}_0(M)}^*$ is weak* continuous; that is, in order to verify weak* continuity it suffices to do so for bounded monotone nets of Lipschitz functions. This solves a problem posed by N. Weaver. As an auxiliary result, we show that the series decomposition developed by N. J. Kalton for functionals in the predual of $\mathrm {Lip}_0(M)$ can be partially extended to ${\mathrm {Lip}_0(M)}^*$ .

We construct a separately continuous function $e:E\times K\rightarrow \{0,1\}$ on the product of a Baire space $E$ and a compact space $K$ such that no restriction of $e$ to any non-meagre Borel set in $E\times K$ is continuous. The function $e$ has no points of joint continuity, and, hence, it provides a negative solution of Talagrand’s problem in Talagrand [Espaces de Baire et espaces de Namioka, Math. Ann.270 (1985), 159–164].

The aim of this paper is to characterize the non-negative functions φ defined on (0,∞) for which the Hausdorff operator

$${\rm {\cal H}}_\varphi f(z) = \int_0^\infty f \left( {\displaystyle{z \over t}} \right)\displaystyle{{\varphi (t)} \over t}{\rm d}t$$

${\rm {\cal H}}_a^p ({\open C}_ + )$

$p\in [1,\infty ]$

In this paper, we investigate Dirichlet spaces ${{D}_{\mu }}$ with superharmonic weights induced by positive Borel measures $\mu $ on the open unit disk. We establish the Alexander-Taylor-Ullman inequality for ${{D}_{\mu }}$ spaces and we characterize the cases where equality occurs. We define a class of weighted Hardy spaces $H_{\mu }^{2}$ via the balayage of the measure $\mu $ . We show that ${{D}_{\mu }}$ is equal to $H_{\mu }^{2}$ if and only if $\mu $ is a Carleson measure for ${{D}_{\mu }}$ . As an application, we obtain the reproducing kernel of ${{D}_{\mu }}$ when $\mu $ is an infinite sum of point-mass measures. We consider the boundary behavior and innerouter factorization of functions in ${{D}_{\mu }}$. We also characterize the boundedness and compactness of composition operators on ${{D}_{\mu }}$.

In this paper, we show that the Möbius invariant function space ${{\mathcal{Q}}_{p}}$ can be generated by variant Dirichlet type spaces ${{\mathcal{D}}_{\mu ,p}}$ induced by finite positive Borel measures $\mu $ on the open unit disk. A criterion for the equality between the space ${{\mathcal{D}}_{\mu ,p}}$ and the usual Dirichlet type space ${{\mathcal{D}}_{p}}$ is given. We obtain a sufficient condition to construct different ${{\mathcal{D}}_{\mu ,p}}$ spaces and provide examples. We establish decomposition theorems for ${{\mathcal{D}}_{\mu ,p}}$ spaces and prove that the non-Hilbert space ${{\mathcal{Q}}_{p}}$ is equal to the intersection of Hilbert spaces ${{\mathcal{D}}_{\mu ,p}}$. As an application of the relation between ${{\mathcal{Q}}_{p}}$ and ${{\mathcal{D}}_{\mu ,p}}$ spaces, we also obtain that there exist different ${{\mathcal{D}}_{\mu ,p}}$ spaces; this is a trick to prove the existence without constructing examples.

The purpose of this paper is to characterize positive measure $\mu$ on the unit disk such that the analytic Morrey space $\mathcal{A}{{\mathcal{L}}_{p,\eta }}$ is boundedly and compactly embedded to the tent space

$$\mathcal{J}_{q,1-\frac{q}{p}\left( 1-\eta \right)}^{\infty }\left( \mu \right)$$

for the case $1\,\le \,q\,\le \,p\,<\,\infty$ respectively. As an application, these results are used to establish the boundedness and compactness of integral operators and multipliers between analytic Morrey spaces.

A set is shy or Haar null (in the sense of Christensen) if there exists a Borel set and a Borel probability measure μ on C[0, 1] such that and for all f ∈ C[0, 1]. The complement of a shy set is called a prevalent set. We say that a set is Haar ambivalent if it is neither shy nor prevalent.

The main goal of the paper is to answer the following question: what can we say about the topological properties of the level sets of the prevalent/non-shy many f ∈ C[0, 1]?

The classical Bruckner–Garg theorem characterizes the level sets of the generic (in the sense of Baire category) f ∈ C[0, 1] from the topological point of view. We prove that the functions f ∈ C[0, 1] for which the same characterization holds form a Haar ambivalent set.

In an earlier paper, Balka et al. proved that the functions f ∈ C[0, 1] for which positively many level sets with respect to the Lebesgue measure λ are singletons form a non-shy set in C[0, 1]. The above result yields that this set is actually Haar ambivalent. Now we prove that the functions f ∈ C[0, 1] for which positively many level sets with respect to the occupation measure λ ◦ f –1 are not perfect form a Haar ambivalent set in C[0, 1].

We show that for the prevalent f ∈ C[0, 1] for the generic y ∈ f([0, 1]) the level set f –1(y) is perfect. Finally, we answer a question of Darji and White by showing that the set of functions f ∈ C[0, 1] for which there exists a perfect set Pf ⊂ [0, 1] such that fʹ(x) = ∞ for all x ∈ Pf is Haar ambivalent.

We study the strong continuity of semigroups of composition operators on local Dirichlet spaces.

In this paper we characterize the isometries of subspaces of the little Zygmund space. We show that the isometries of these spaces are surjective and represented as integral operators. We also show that all hermitian operators on these settings are bounded

We point out how the “Fundamental Theorem of Stability Theory”, namely the equivalence between the “non order property” and definability of types, proved by Shelah in the 1970s, is in fact an immediate consequence of Grothendieck’s “Critères de compacité” from 1952. The familiar forms for the defining formulae then follow using Mazur’s Lemma regarding weak convergence in Banach spaces.

Let $X$ and $Y$ be metric spaces and $E$, $F$ be Banach spaces. Suppose that both $X$ and $Y$ are realcompact, or both $E$, $F$ are realcompact. The zero set of a vector-valued function $f$ is denoted by $z\left( f \right)$. A linear bijection $T$ between local or generalized Lipschitz vector-valued function spaces is said to preserve zero-set containments or nonvanishing functions if

$$z\left( f \right)\,\subseteq \,z\left( g \right)\,\,\,\,\Leftrightarrow \,\,\,\,z\left( Tf \right)\,\subseteq \,z\left( Tg \right),\,\,\,\,\,\text{or}\,\,\,\,z\left( f \right)\,=\,\varnothing \,\,\,\Leftrightarrow \,\,\,z\left( Tf \right)\,=\,\varnothing ,$$

respectively. Every zero-set containment preserver, and every nonvanishing function preserver when $\dim\,E\,=\,\dim\,F\,<\,+\infty$, is a weighted composition operator $\left( Tf \right)\left( y \right)\,=\,{{J}_{y}}\left( f\left( \tau \left( y \right) \right) \right)$. We show that the map $\tau \,:\,Y\,\to \,X$ is a locally (little) Lipschitz homeomorphism.

We prove that for a large class of Banach function spaces continuity and holomorphy of superposition operators are equivalent and that bounded superposition operators are continuous. We also use techniques from infinite dimensional holomorphy to establish the boundedness of certain superposition operators. Finally, we apply our results to the study of superposition operators on weighted spaces of holomorphic functions and the $F(p, \alpha , \beta )$ spaces of Zhao. Some independent properties on these spaces are also obtained.

We show that for a normal locally-$\mathscr{P}$ space $X$ (where $\mathscr{P}$ is a topological property subject to some mild requirements) the subset ${C}_{\mathscr{P}} (X)$ of ${C}_{b} (X)$ consisting of those elements whose support has a neighborhood with $\mathscr{P}$, is a subalgebra of ${C}_{b} (X)$ isometrically isomorphic to ${C}_{c} (Y)$ for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$ is explicitly constructed as a subspace of the Stone–Čech compactification $\beta X$ of $X$ and contains $X$ as a dense subspace. Under certain conditions, ${C}_{\mathscr{P}} (X)$ coincides with the set of those elements of ${C}_{b} (X)$ whose support has $\mathscr{P}$, it moreover becomes a Banach algebra, and simultaneously, $Y$ satisfies ${C}_{c} (Y)= {C}_{0} (Y)$. This includes the cases when $\mathscr{P}$ is the Lindelöf property and $X$ is either a locally compact paracompact space or a locally-$\mathscr{P}$ metrizable space. In either of the latter cases, if $X$ is non-$\mathscr{P}$, then $Y$ is nonnormal and ${C}_{\mathscr{P}} (X)$ fits properly between ${C}_{0} (X)$ and ${C}_{b} (X)$; even more, we can fit a chain of ideals of certain length between ${C}_{0} (X)$ and ${C}_{b} (X)$. The known construction of $Y$ enables us to derive a few further properties of either ${C}_{\mathscr{P}} (X)$ or $Y$. Specifically, when $\mathscr{P}$ is the Lindelöf property and $X$ is a locally-$\mathscr{P}$ metrizable space, we show that

$$\begin{eqnarray*}\dim C_{\mathscr{P}}(X)= \ell \mathop{(X)}\nolimits ^{{\aleph }_{0} } ,\end{eqnarray*}$$

$\ell (X)$

$X$

$\mathscr{P}$

$X$

$$\begin{eqnarray*}Y= {\mathrm{int} }_{\beta X} \upsilon X\end{eqnarray*}$$

$\upsilon X$

$X$