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The purpose of this work is to prove the non-triviality of anisotropic Roumieu Gelfand–Shilov spaces S { N } { M } ⁢ (ℝ n) , and to establish the inclusion between them. [ABSTRACT FROM AUTHOR]

Influenced by the concept of a p-compact operator due to Sinha and Karn (Stud Math 150(1): 17–33, 2002), we introduce p-compact Bloch maps of the open unit disk D ⊆ C to a complex Banach space X, and obtain its most outstanding properties: surjective Banach ideal property, Möbius invariance, linearisation on the Bloch-free Banach space over D , inclusion properties, factorisation of their derivatives, and transposition on the normalized Bloch space. We also present right p-nuclear Bloch maps of D to X and study its relation with p-compact Bloch maps. [ABSTRACT FROM AUTHOR]

We study several variants of 2-local isometries (or algebra isomorphisms) on some function algebras, e.g., Lipschitz algebras, algebras of differential functions, algebras of absolutely continuous functions and algebras of continuous functions with bounded variation. A typical result is this: if ϕ is surjective map between function algebra mentioned above with the property that for any pair f, g there is an algebra isomorphism ϕ f , g such that ϕ (f) ϕ (g) = ϕ f , g (f g) , then ϕ can be written as a weighted composition operator. [ABSTRACT FROM AUTHOR]

We prove that if A is an infinite Boolean algebra in the ground model V and P is a notion of forcing adding any of the following reals: a Cohen real, an unsplit real, or a random real, then, in any P -generic extension V[G], A has neither the Nikodym property nor the Grothendieck property. A similar result is also proved for a dominating real and the Nikodym property. [ABSTRACT FROM AUTHOR]

We study sufficient conditions on weight functions under which norm approximations by analytic polynomials are possible. The weights we study include radial, non-radial, and angular weights. [ABSTRACT FROM AUTHOR]

Let Q, K be connected open subsets of R m and A(X), A(Y) be some kind of function spaces. We will study the 2-local isometries between the vector-valued differentiable function spaces C 0 p (Q , A (X)) and C 0 p (K , A (Y)) , and show that they can be written as weighted composition operators. [ABSTRACT FROM AUTHOR]

We study the Korenblum Maximum Principle for various weighted Fock spaces. The main tool relies on applying special cases of Ramanujan's Master Theorem involving the Gamma function and the Mellin transform of the Dirichlet series. It is interesting that with elementary probe functions, we still can obtain closed-form upper bound expressions, in terms of several well-known special functions, for the Korenblum constants of various weighted Fock spaces. For the first time, we obtain upper bounds of the Korenblum constant for the finite and infinite intersections of weighted Fock spaces and proved that the Korenblum Maximum Principle fails for those infinite intersections of weighted Fock spaces. Some open questions are provided. [ABSTRACT FROM AUTHOR]

We study the weighted Bergman spaces on finitely connected planar domains. They are isomorphic to the product of weighted Bergman spaces on the unit disk. Using this, we characterize the Carleson embeddings and prove kernel estimates. We characterize bounded, compact and Schatten class composition and Toeplitz operators on these spaces. Our results generalize several recent ones in the unit disk or simply connected case. [ABSTRACT FROM AUTHOR]

As a continuation of the study of polynomial approximation and composition operators on Dirichlet spaces of unit disk, which has settled a problem posed by Cima in 1976, the present paper aims to consider the case of the unbounded domains, such as the half-plane. Specifically, we may obtain the rational approximations in the Dirichlet spaces and characterize the composition operators which have a dense range on the Dirichlet spaces over the half-plane. Moreover, this paper also considers the relationship between the Dirichlet spaces and Hardy spaces on half-plane. [ABSTRACT FROM AUTHOR]

In this article, we construct the fractal interpolation functions (FIFs) on the Sierpiński gasket (SG) by taking data set at the n th level. We discuss the oscillation space, L q space and some other function spaces on SG , and determine some conditions under which, this FIF belongs to these spaces. We obtain the fractal dimension of the graph of the FIF and the Hausdorff dimension of the invariant measure supported on the graph of the FIF. We also prove that this FIF has finite energy. After that, we discuss the restrictions of the FIF on the bottom line segment of SG and prove that the restriction is again an FIF on the bottom line. We estimate the fractal dimension of the graph of the Riemann-Liouville fractional integral of the restrictions of the FIF on the bottom line segment of SG. Lastly, we prove that the Hausdorff and box-counting dimension of the graph of the restriction of any harmonic function on the bottom line segment of SG are 1 and for the Riemann- Liouville fractional integral of this restriction function, these dimensions are also 1. [ABSTRACT FROM AUTHOR]

Let Lip (I) be the Banach algebra of all Lipschitz functions on the closed unit interval I with the norm ‖ f ‖ L = ‖ f ‖ ∞ + L (f) for f ∈ Lip (I) , where L(f) is the Lipschitz constant of f. We denote by C 1 (I , Lip (I)) the Banach algebra of all continuously differentiable functions F from I to Lip (I) equipped with the norm ‖ F ‖ Σ = sup s ∈ I ‖ F (s) ‖ L + sup t ∈ I ‖ D (F) (t) ‖ L for F ∈ C 1 (I , Lip (I)) . In this paper, we prove that if T is a surjective, not necessarily linear, isometry on C 1 (I , Lip (I)) , then T - T (0) is a weighted composition operator or its complex conjugation. Among other things, any surjective complex linear isometry on C 1 (I , Lip (I)) is of the following form: c 1 F (τ 1 (s) , τ 2 (x)) , where c 1 is a complex number of modulus 1, and τ 1 and τ 2 are isometries of I onto itself. [ABSTRACT FROM AUTHOR]

The paper studies families of two-parameter Bergman type operators T β , λ , S β , λ , Φ β , δ in mixed norm and Besov spaces on the unit ball of R n. Motivated by a series of papers by Choe et al., we extend estimation theorems for harmonic reproducing kernels. This enables us to obtain boundedness of operators T β , λ , S β , λ , Φ β , δ on mixed norm and Besov spaces for appropriate parameters. A necessary and sufficient condition is found for T β , λ and S β , λ to be bounded on mixed norm spaces L (p , q , α). For non-positive α , Bergman projection T β , 0 continuously maps mixed norm space L (p , q , α) onto corresponding Besov space. Furthermore, a new operator Φ β , δ of Bergman type is found that projects Besov space of smooth functions onto its harmonic subspace. [ABSTRACT FROM AUTHOR]

In this note, we give the mixing, and therefore hypercyclic, property of the exponential of the differential operator on Banach spaces related to Fock spaces. Besides, we also characterize the mixing property of translation operators on such spaces and discuss the different behaviors of these operators between these spaces and classical Fock spaces. [ABSTRACT FROM AUTHOR]

We introduce the strong Gelfand–Phillips property for locally convex spaces and give several characterizations of this property. We characterize the strong Gelfand–Phillips property among locally convex spaces admitting a stronger Banach space topology. If C T (X) is a space of continuous functions on a Tychonoff space X, endowed with a locally convex topology T between the pointwise topology and the compact-open topology, then: (a) the space C T (X) has the strong Gelfand–Phillips property iff X contains a compact countable subspace K ⊆ X of finite scattered height such that for every functionally bounded set B ⊆ X the complement B \ K is finite, (b) the subspace C T b (X) of C T (X) consisting of all bounded functions on X has the strong Gelfand–Phillips property iff X is a compact countable space of finite scattered height. [ABSTRACT FROM AUTHOR]

Srivastava and Singh [9] generalized the idea of Λ -strong convergence given by Móricz [6] to T-strong convergence of numerical sequences and Fourier series. In this present work, we introduce the idea of Tr-strong convergence r ∈ N , by generalizing the idea of T-strong convergence. This paper also generalizes some results of Kórus [5], as shown in conclusion. Additionally, we provide a relationship between T-strong convergence, T2-strong convergence, ⋯ , Tr-strong convergence, and ordinary convergence. We also demonstrate that this idea can be used to explain the Tr-convergence of Fourier series in the C-metric and the Lp-metric. [ABSTRACT FROM AUTHOR]

Using Mujica’s linearization theorem, we extend to the holomorphic setting some classical characterizations of compact (weakly compact, Rosenthal, Asplund) linear operators between Banach spaces such as the Schauder, Gantmacher and Gantmacher–Nakamura theorems and the Davis–Figiel–Johnson–Pełczynski, Rosenthal and Asplund factorization theorems. [ABSTRACT FROM AUTHOR]

Consider the space of weighted binomial matrices in the Nakano sequence space of soft functions. We have offered some geometric and topological structures of the multiplication operator acting on this space and its associated operator ideal. The existence of a fixed point of the Kannan contraction operator in this prequasioperator ideal is confirmed. Finally, we discuss many applications of solutions to nonlinear stochastic dynamical matrix systems and illustrative examples of our findings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the 2-local isometries between the vector-valued differentiable function spaces C 0 p (X , E) and C 0 q (Y , E) , where X, Y are open subsets of R and E is a smooth, strictly convex, reflexive and 2-iso-reflexive Banach space (in particular, when 1 ≤ p < ∞ and p ≠ 2 , ℓ p satisfies such conditions). We show that such maps can be written as weighted composition operators. [ABSTRACT FROM AUTHOR]

Recently, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBS) for applied and theoretical purposes such as machine learning, sampling reconstruction, sparse approximation and functional analysis. Existing constructions include the reflexive RKBS via a bilinear form, the semi-inner-product RKBS, the RKBS with ℓ1 norm, the p-norm RKBS via generalized Mercer kernels, etc. The definitions of RKBS and the associated reproducing kernel in those references are dependent on the construction. Moreover, relations among those constructions are unclear. We explore a generic definition of RKBS and the reproducing kernel for RKBS that is independent of construction. Furthermore, we propose a framework of constructing RKBSs that leads to new RKBSs based on Orlicz spaces and unifies existing constructions mentioned above via a continuous bilinear form and a pair of feature maps. Finally, we develop representer theorems for machine learning in RKBSs constructed in our framework, which also unifies representer theorems in existing RKBSs. [ABSTRACT FROM AUTHOR]

In the present work, we investigate the dynamic behavior of composition operators on Hilbert spaces of Dirichlet series. Especially, we characterize completely their cyclicity. Then it is shown that there are no hypercyclic composition operators, and given a certain class of operators whose adjoints are hypercyclic. All these indicate that there are indigent cyclic composition operators. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the projective limit of a certain class of Dirichlet-type spaces is isomorphic to the Bloch-type spaces B α . We provide a complete characterization of bounded or compact embeddings of B α into a family of Lebesgue spaces by the derivative operator. As an application, we prove precise necessary and sufficient criteria for boundedness/compactness of the generalized composition operators. The Dirichlet-type intermediate spaces constructed in this paper serve very useful. [ABSTRACT FROM AUTHOR]

Following the ideas of recent research in Karapetyants and Samko [Mixed norm variable exponent Bergman space on the unit disc. Complex Var Elliptic Equ. 2016;61:1090–1106], we continue the study of new weighted spaces of holomorphic functions on the unit disc with the mixed norm defined in terms of conditions on Fourier coefficients of a function. Here we present a general approach for the weighted case. We study the boundedness of the weighted Bergman projection and discuss boundedness and compactness of Toeplitz operators with radial symbols. We also provide a characterization of functions in these spaces. [ABSTRACT FROM AUTHOR]

Some topological and geometric behavior of the space of all sequences whose generalized mean transforms are in Nakano sequence space, the multiplication mappings acting on it, and the eigenvalue distribution of mappings ideal generated by this space and s-numbers are discussed. We construct the existence of a fixed point of Kannan contraction mapping on these spaces. Several numerical experiments are presented to illustrate our results. Moreover, some successful applications to the existence of solutions of nonlinear difference equations are explained. The strength here is that the current results are constructed under flexible setups given by controlling the weight and power of these spaces. [ABSTRACT FROM AUTHOR]

We study the natural function space for infinitely wide two-layer neural networks with ReLU activation (Barron space) and establish different representation formulae. In two cases, we describe the space explicitly up to isomorphism. Using a convenient representation, we study the pointwise properties of two-layer networks and show that functions whose singular set is fractal or curved (for example distance functions from smooth submanifolds) cannot be represented by infinitely wide two-layer networks with finite path-norm. We use this structure theorem to show that the only C 1 -diffeomorphisms which preserve Barron space are affine. Furthermore, we show that every Barron function can be decomposed as the sum of a bounded and a positively one-homogeneous function and that there exist Barron functions which decay rapidly at infinity and are globally Lebesgue-integrable. This result suggests that two-layer neural networks may be able to approximate a greater variety of functions than commonly believed. [ABSTRACT FROM AUTHOR]

In this paper, we continue the study of new weighted spaces of holomorphic functions on the unit disc with the mixed norm defined in terms of conditions on Fourier coefficients of a function. Here we present the case in which the corresponding conditions are related to grand and small Lebesgues spaces, i.e. the Fourier coefficients as functions of radial variable belong to either grand or small space, and the norms of these coefficients taken in the corresponding space all together form a l q sequence. We provide the characterization of functions in such mixed norm spaces and study the boundedness of the holomorphic projection. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear Schrödinger equation involving the fractional Laplacian with Dirichlet condition: (- Δ) α 2 u + A (x) u = f (x , u , ∇ u) in Ω , u > 0 , in Ω ; u ≡ 0 , in R n \ Ω , where Ω is a domain (bounded or unbounded) in R n which is convex in x 1 -direction. By using some ideas of maximum principle and the direct moving plane method, we prove that the solutions are strictly increasing in x 1 -direction in the left half domain of Ω . Symmetry of some solutions are also proved. Meanwhile, we obtain a Liouville type theorem on the half space R + n . [ABSTRACT FROM AUTHOR]

Let X, Y be two w ∗ -almost smooth Banach spaces, C (B (X ∗) , w ∗) be the Banach space of all continuous real-valued functions on B (X ∗) endowed with the supremum norm and C + (B (X ∗) , w ∗) be the positive cone of C (B (X ∗) , w ∗) . In this paper, we show that if F : C + (B (X ∗) , w ∗) → C + (B (Y ∗) , w ∗) is a standard almost surjective ε -isometry, then there exists a homeomorphism τ : B (X ∗) → B (Y ∗) in the w ∗ -topology such that for any x ∗ ∈ B (X ∗) , we have | ⟨ δ x ∗ , f ⟩ - ⟨ δ τ (x ∗) , F (f) ⟩ | ≤ 2 ε , for all f ∈ C + (B (X ∗) , w ∗). As its application, we show that if U : C (B (X ∗) , w ∗) → C (B (Y ∗) , w ∗) is the canonical linear surjective isometry induced by the homeomorphism γ = τ - 1 : B (Y ∗) → B (X ∗) in the w ∗ -topology, then ‖ F (f) - U (f) ‖ ≤ 2 ε , for all f ∈ C + (B (X ∗) , w ∗). [ABSTRACT FROM AUTHOR]

Let u be an analytic function in the unit disk D and φ be an analytic self-map of D . We give characterizations of the symbols u and φ for which the multiplication operator M u and the weighted composition operator M u , φ are isometries of BMOA. [ABSTRACT FROM AUTHOR]

This paper concerns univalent analytic functions f for which log ⁡ f ′ belong to Dirichlet-Morrey spaces. We establish some characterizations of Dirichlet-Morrey domains. Our results involve Schwarzian derivatives, quasiconformal extensions and Grunsky kernels of univalent functions. [ABSTRACT FROM AUTHOR]

Let G be a locally compact second countable Hausdorff space with a positive regular Borel measure and ω is a weight on G. In this article, we provide necessary and sufficient conditions for the hypercyclic weighted translations acting on the weighted space L p (G , ω) (1 ≤ p < ∞) in two different cases. Also, some examples are given to illustrate that the results in the first case generalize the characterizations on hypercyclicity for unilateral weighted shifts studied by Salas [16], and the results in the second case generalize Chen and Chu's work in [8]. Furthermore, we give characterizations of hypercyclicity for adjoint operators of these weighted translations. [ABSTRACT FROM AUTHOR]

We have constructed the sequence space (Ξ (ζ , t)) υ , where ζ = (ζ l) is a strictly increasing sequence of positive reals tending to infinity and t = (t l) is a sequence of positive reals with 1 ≤ t l < ∞ , by the domain of (ζ l) -Cesàro matrix in the Nakano sequence space ℓ (t l) equipped with the function υ (f) = ∑ l = 0 ∞ (| ∑ z = 0 l f z Δ ζ z | ζ l) t l for all f = (f z) ∈ Ξ (ζ , t) . Some geometric and topological properties of this sequence space, the multiplication mappings defined on it, and the eigenvalues distribution of operator ideal with s-numbers belonging to this sequence space have been investigated. The existence of a fixed point of a Kannan pre-quasi norm contraction mapping on this sequence space and on its pre-quasi operator ideal formed by (Ξ (ζ , t)) υ and s-numbers is presented. Finally, we explain our results by some illustrative examples and applications to the existence of solutions of nonlinear difference equations. [ABSTRACT FROM AUTHOR]

In this article, we investigate the notion of the pre-quasi norm on a generalized Cesàro backward difference sequence space of non-absolute type (Ξ (Δ , r)) ψ under definite function ψ. We introduce the sufficient set-up on it to form a pre-quasi Banach and a closed special space of sequences (sss), the actuality of a fixed point of a Kannan pre-quasi norm contraction mapping on (Ξ (Δ , r)) ψ , it supports the property (R) and has the pre-quasi normal structure property. The existence of a fixed point of the Kannan pre-quasi norm nonexpansive mapping on (Ξ (Δ , r)) ψ and the Kannan pre-quasi norm contraction mapping in the pre-quasi Banach operator ideal constructed by (Ξ (Δ , r)) ψ and s-numbers has been determined. Finally, we support our results by some applications to the existence of solutions of summable equations and illustrative examples. [ABSTRACT FROM AUTHOR]

This paper considers the problem of restricting the short-time Fourier transform to sets of nonzero measure in the plane. Thereby, we study under which conditions one has a sampling set and provide estimates of the corresponding sampling bound. In particular, we give a quantitative estimate for the lower sampling bound in the case of Hermite windows and derive a sufficient condition for a large class of windows in terms of a certain planar density. On the way, we prove a Remez-type inequality for polyanalytic functions. [ABSTRACT FROM AUTHOR]

We suggest a sufficient setting on any linear space of sequences V such that the class B V s of all bounded linear mappings between two arbitrary Banach spaces with the sequence of s-numbers in V constructs a map ideal. We define a new sequence space (ces r 1 , r 2 t) υ for definite functional υ by the domain of (r 1 , r 2) -Cesàro matrix in ℓ t , where r 1 , r 2 ∈ (0 , ∞) and 1 ≤ t < ∞ . We examine some geometric and topological properties of the multiplication mappings on (ces r 1 , r 2 t) υ and the pre-quasi ideal B (ces r 1 , r 2 t) υ s . [ABSTRACT FROM AUTHOR]

Let H (B) denote the space of holomorphic functions on the unit ball B of C n , n ∈ N , ψ , ϕ ∈ H (B) , ϕ (B) ⊂ B . Let C ϕ , M ψ and R denote the composition, multiplication and radial derivative operators, respectively. To treat the operators induced by products of these operators in a unified manner, we study the operator T ψ 1 , ψ 2 , ψ 3 , ϕ = M ψ 1 C ϕ + M ψ 2 C ϕ R + M ψ 3 R C ϕ . We characterize the boundedness, compactness and order boundedness of T ψ 1 , ψ 2 , ψ 3 , ϕ mapping from a large class of Banach spaces X of holomorphic functions into the weighted-type space H μ ∞ (or H μ , 0 ∞ ), and give its norm estimate and essential norm estimate. We also obtain the above characterizations and estimates for the weighted composition operator M ψ C ϕ mapping from X into the Bloch-type space B μ (or B μ , 0 ). Our results show that the above characterizations and estimates of these operators depend only on the symbols and the norm of the point-evaluation functionals on the domain space. [ABSTRACT FROM AUTHOR]

Let X and Y be compact Hausdorff spaces and let F : C (X) → C (Y) be a (non-surjective) coarse (M , ε) -quasi-isometry with M < 8 / 7 . Assume also that the range of F "approximates" C(Y) well, then X and Y are homeomorphic. Moreover, if U : C (X) → C (Y) is the canonical isometry induced by the homeomorphism, then ‖ F (f) - M U (f) ‖ ≤ 8 M 2 - 1 M ‖ f ‖ + Δ ε for every f ∈ C (K) , where Δ depends only on M and the degree of approximation of C(Y) by the range of F. (See the Introduction for the precise technical approximation condition and the estimate of Δ ). The approximation result is new even for non-surjective isometries. [ABSTRACT FROM AUTHOR]

We study harmonic Besov spaces b α p on the unit ball of R n , where 0

The purpose of the paper is to study the coefficient multipliers of the Hardy spaces H p associated with Jacobi expansions of exponential type. The main results are about the boundedness from H p to ℓ q of the multiplier operators in terms of Jacobi expansions of exponential type for (i) p = 1 , 2 ≤ q < ∞ ; (ii) γ (α , β) − 1 < p < 1 ≤ q < ∞ , under appropriate conditions, where γ (α , β) ∈ (1 , ∞ ] is a number depending on the parameters of the Jacobi system. [ABSTRACT FROM AUTHOR]

In earlier papers we introduced and studied the notion of fractal operator that maps a continuous scalar valued function on a compact interval in R to its fractal (self-referential) analogue. Further, it has been observed that by perturbing known Schauder bases via the bijective bicontinuous fractal operator, Schauder bases consisting of self-referential functions can be constructed for standard function spaces. Motivated by the theory and applications of the fractal operator, in this note we propose a new kind of perturbation of linear operators. We consider a pair of linear operators L and T on a Banach space X such that ‖ T x - x ‖ ≤ k ‖ L x - x ‖ for all x ∈ X and some k > 0 . We establish that the property of being bounded below and that of having a closed range are stable under the perturbation considered herein. As an immediate application, it is deduced that a given Schauder basis in X can be transformed to Schauder sequences. Perturbation of linear operators via an equation of the form ‖ T x - x ‖ = k ‖ T x - L x ‖ is also alluded and its application in constructing new Schauder bases from a given one is considered. [ABSTRACT FROM AUTHOR]

The objective of this article is to establish a fixed point theorem using different class of control functions, which is a generalization of Darbo's fixed point theorem (DFPT) and its various versions obtained by several authors. Using this result we obtain existence of at least one positive solution for a quadratic integral equation of Fredholm type. [ABSTRACT FROM AUTHOR]

In this paper, we characterize disjoint hypercyclic powers of weighted pseudo-shifts on an arbitrary F-sequence space. As a special case, we deduce the equivalent conditions for the disjoint hypercyclicity of finitely many different powers of weighted shifts on ℓ 2 (Z , K) with weight sequence { A n } n = - ∞ ∞ of positive invertible diagonal operators on K . [ABSTRACT FROM AUTHOR]

We establish a distance formula from a Bloch-type function to , which gives a characterization of the closure of in the Bloch-type space and generalizes some results on the distance formulas from a Bloch function to BMOA, and spaces, respectively. Our proof presented here is based on several estimates on the weight function K, which are of some independent interest. [ABSTRACT FROM AUTHOR]

In this paper, we consider the characterization of norm–parallelism problem in some classical Banach spaces. In particular, for two continuous functions f, g on a compact Hausdorff space K, we show that f is norm–parallel to g if and only if there exists a probability measure (i.e., positive and of full measure equal to 1) μ with its support contained in the norm-attaining set { x ∈ K : | f (x) | = ‖ f ‖ } such that | ∫ K f (x) ¯ g (x) d μ (x) | = ‖ f ‖ ‖ g ‖ . [ABSTRACT FROM AUTHOR]

Abstract: We describe exactly and fully which of the spaces of holomorphic functions in the title are included in which others. We provide either new results or new proofs. More importantly, we construct explicit functions in each space that show our relations are strict and the best possible. Many of our inclusions turn out to be sharper than the Sobolev imbeddings. [ABSTRACT FROM AUTHOR]

Let w be an unbounded radial weight on the complex plane. We study the following approximation problem: find a proper holomorphic map f:C→Cn such that |f| is equivalent to w. We give several characterizations of those w for which the problem is solvable. In particular, a constructive characterization is given in terms of tropical power series. Moreover, the following natural objects and properties are involved: essential weights on the complex plane, approximation by power series with positive coefficients, and approximation by the maximum of a holomorphic function modulus. Extensions to several complex variables and approximation by harmonic maps are also considered. [ABSTRACT FROM AUTHOR]

A Banach space X has Pełczyński's property (V) if for every Banach space Y every unconditionally converging operator [ABSTRACT FROM AUTHOR]

We prove Banach–Stone type theorems for linear isometries on spaces of continuous sections of Banach space bundles. These extend a previous result for vector-valued continuous function spaces. Two common methods, examining the extreme point sets of the dual spaces and examining peak points, are adopted to prove two similar but slightly different theorems. [ABSTRACT FROM AUTHOR]

We study surjective isometries between subspaces of continuous functions containing all constant functions and separating the points of the underlying spaces. In many contexts, every such isometry is represented by a combination of a weighted composition operator and its complex conjugate, called the canonical form, while there exists an isometry which does not take such a form ( [14] ). We seek a topological condition on compact Hausdorff spaces such that every surjective isometry on function spaces over the spaces has the canonical form. Also we extend the construction of [14] to show that, if a compact metrizable space X admits a semi-free action of the circle group with a global section, then there exists an isometry of a function space on X which does not take the canonical form. [ABSTRACT FROM AUTHOR]

Let ( M 1 , d 1 ) , ( M 2 , d 2 ) be metric spaces. A map f : M 1 → M 2 is said to be locally open at an x 1 ∈ M 1 , if for every ε > 0 one finds a δ > 0 such that B ( f ( x 1 ) , δ ) ⊂ f ( B ( x 1 , ε ) ) ; here B ( x , r ) stands for the closed ball with center x and radius r . We are particularly interested in the following special case: X , Y , Z are normed spaces, the spaces L ( X , Y ) , L ( Y , Z ) , L ( X , Z ) of linear continuous operators are provided with the operator norm, and the map under consideration is the bilinear map ( S , T ) ↦ S ∘ T (from L ( Y , Z ) × ( L ( X , Y ) to L ( X , Z ) ). For which pairs ( S 0 , T 0 ) ∈ L ( Y , Z ) × ( L ( X , Y ) is it locally open? The main result of the paper gives a complete characterization of pairs ( S , T ) at which this map is locally open in the case of finite-dimensional spaces X , Y , Z . [ABSTRACT FROM AUTHOR]

Letbe a bounded domain with-smooth boundary, whereis a Dini-smooth modulus of continuity. We prove that the restricted Beurling transformis bounded on the Campanato space. [ABSTRACT FROM AUTHOR]