On an integral operator between Bloch-type spaces on the unit ball

AbstractWe characterize the boundedness and compactness of the following integral-type operatorIφg(f)(z)=∫01Rf(φ(tz))g(tz)dtt,z∈B, where g is a holomorphic function on the unit ball B⊂Cn such that g(0)=0, and φ is a holomorphic self-map of B, acting from α-Bloch spaces to Bloch-type spaces on B

Cesàro-type operators associated with Borel measures on the unit disc acting on some Hilbert spaces of analytic functions

Given a complex Borel measure μon the unit disc D={z∈C:|z| <1}, we consider the Cesàro-type operator Cμdefined on the space Hol(D)of all analytic functions in Das follows: If f∈Hol(D), f(z) = ∞n=0anzn(z∈D), then Cμ(f)(z) = ∞n=0μn nk=0ak zn, (z∈D), where, for n ≥0, μndenotes the n-th moment of the measure μ, that is, μn= Dwndμ(w). We study the action of the operators Cμon some Hilbert spaces of analytic function in D, namely, the Hardy space H2and the weighted Bergman spaces A2α(α >−1). Among other results, we prove that, if we set Fμ(z) = ∞n=0μnzn(z∈D), then Cμis bounded on H2or on A2αif and only if Fμbelongs to the mean Lipschitz space Λ21/2. We prove also that Cμis a Hilbert-Schmidt operator on H2if and only if Fμbelongs to the Dirichlet space D, and that Cμis a Hilbert-Schmidt operator on A2αif and only if Fμbelongs to the Dirichlet-type space D2−1−α.Funding for open access charge: Universidad de Málaga / CBUA This research is supported in part by a grant from “El Ministerio de Economía y Competitividad” Spain (PGC2018-096166-B-I00) and by grants from la Junta de Andalucía (FQM-210 and UMA18-FEDERJA-002)

Composition operators from the weighted . . .

The boundedness of the composition operator from the weighted Bergman space to the recently introduced by the author, the nth weighted space on the unit disc, is characterized. Moreover, the norm of the operator in terms of the inducing function and weights is estimated

Weighted composition operators from . . .

Let H B denote the space of all holomorphic functions on the unit ball B. Let u ∈ H B and ϕ be a holomorphic self-map of B. In this paper, we investigate the boundedness and compactness of the weighted composition operator uC ϕ from the general function space F p, q, s to the weightedtype space H ∞ μ in the unit ball

Weighted composition operators and weighted conformal invariance

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de Lectura: 27-07-2021En esta tesis tratamos ciertos problemas relacionados con los operadores de composici´on ponderados. Estudiamos c´omo act´uan estos operadores en espacios de funciones anal´ıticas en D o en un dominio acotado Ω ⊂ C. En primer lugar nos centramos en una familia amplia de espacios de Hilbert de funciones anal´ıticas en el disco unidad, los cuales satisfacen solamente un n´umero reducido de axiomas y cuyo n´ ucleo reproductor tiene la forma usual. A estos espacios se les llama espacios de Hardy con peso. En estos espacios caracterizamos los operadores de composici´on ponderados que son co-isom´etricos (equivalentemente, unitarios). El resultado principal nos revela una dicotom´ıa al identificar una familia especifica de espacios de Hardy con peso como los ´unicos espacios en los cuales existen operadores no triviales de este tipo. La segunda parte de la tesis est´a dedicada a explorar una clase de espacios de funciones anal´ıticas los cuales comparten una cierta propiedad de invariancia conforme ponderada. Para ser m´as preciso, en esta parte presentamos una aproximaci´on general a los espacios que son invariantes bajo los operadores Wϕα, definidos por Wϕαf =(ϕ')α(f ◦ ϕ) con α> 0 y ϕ ∈ Aut(D). Podemos observar que muchos de los espacios de Banach de funciones anal´ıticas cl´asicos como los espacios de crecimiento de Korenblum, los espacios de Hardy, los espacios de Bergman con peso y ciertos espacios de Besov son invariantes bajo estos operadores. Entre otras cosas, en esta parte identificamos el espacio m´as grande, el m´as peque˜no y el “´unico” espacio de Hilbert que satisface esta propiedad de invariancia ponderada para un α> 0 dado. En la ´ultima parte consideramos espacios de Banach abstractos de funciones anal´ıticas en un dominio acotado general los cuales s´olo satisfacen unos pocos axiomas. A continuaci´on ponderados invertibles on, describimos todos los operados de composici´(equivalentemente, sobreyectivos) que act´uan sobre estos espaciosThis thesis treats a number of problems related to weighted composition operators. We study how these operators act on the spaces of analytic functions in D or in a bounded domain Ω ⊂ C. We first focus on a large family of Hilbert spaces of analytic functions in the unit disc which satisfy only a minimum number of axioms and whose reproducing kernels have the usual natural form. These spaces are called weighted Hardy spaces. In these spaces, we characterize the weighted composition operators which are co-isometric (equivalently, unitary). The main result reveals a dichotomy identifying a specific family of weighted Hardy spaces as the only ones that support non-trivial operators of this kind. The second part of the thesis is devoted to exploring a class of spaces of analytic functions which share certain weighted invariant property. More precisely, in this part we present a general approach to the spaces which are invariant under the operators Wϕα, defined by Wϕαf =(ϕ ')α(f ◦ ϕ) with α> 0 and ϕ ∈ Aut(D). We observe that many common examples of Banach spaces of analytic functions like Korenblum growth classes, Hardy spaces, standard weighted Bergman and certain Besov spaces are invariant under these operators. Among other things, we identify the largest and the smallest as well as the “unique” Hilbert space satisfying this weighted invariant property for a given α> 0. In the last part, we consider abstract Banach spaces of analytic functions on general bounded domains that satisfy only a minimum number of axioms. Then, we describe all invertible (equivalently, surjective) weighted composition operators acting on such sp

Carleson measures for Besov spaces on the ball with applications

Carleson and vanishing Carleson measures for Besov spaces on the unit ball of C N are characterized in terms of Berezin transforms and Bergman-metric balls. The measures are defined via natural imbeddings of Besov spaces into Lebesgue classes by certain combinations of radial derivatives. Membership in Schatten classes of the imbeddings is considered too. Some Carleson measures are not finite, but the results extend and provide new insight to those known for weighted Bergman spaces. Special cases pertain to Arveson and Dirichlet spaces, and a unified view with the usual Hardy-space Carleson measures is presented by letting the order of the radial derivatives tend to 0. Weak convergence in Besov spaces is also characterized, and weakly 0-convergent families are exhibited. Applications are given to separated sequences, operators of Forelli-Rudin type, gap series, characterizations of weighted Bloch, Lipschitz, and growth spaces, inequalities of Fejér-Riesz and Hardy-Littlewood type, and integration operators of Cesàro type