Multipliers and integration operators between conformally invariant spaces
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Multipliers and integration operators between conformally invariant spaces Daniel Girela1
· Noel Merchán2
Received: 4 November 2019 / Accepted: 5 August 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract In this paper we are concerned with two classes of conformally invariant spaces of analytic functions in the unit disc D, the Besov spaces B p (1 ≤ p < ∞) and the Q s spaces (0 < s < ∞). Our main objective is to characterize for a given pair (X , Y ) of spaces in these classes, the space of pointwise multipliers M(X , Y ), as well as to study the related questions of obtaining characterizations of those g analytic in D such that the Volterra operator Tg or the companion operator Ig with symbol g is a bounded operator from X into Y . Keywords Möbius invariant spaces · Besov spaces · Q s spaces · Multipliers · Integration operators · Carleson measures Mathematics Subject Classification 30H25 · 47B38
1 Introduction Let D = {z ∈ C : |z| < 1} denote the open unit disc of the complex plane C and let Hol(D) be the space of all analytic functions in D endowed with the topology of uniform convergence on compact subsets.
This research is supported in part by a grant from “El Ministerio de Ciencia, Innovación y Universidades”, Spain (PGC2018-096166-B-I00) and by grants from la Junta de Andalucía (FQM-210 and UMA18-FEDERJA-002).
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Daniel Girela [email protected] Noel Merchán [email protected]
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Departamento de Análisis Matemático, Estadística e Investigación Operativa, y Matemática Aplicada, Universidad de Málaga, 29071 Málaga, Spain
2
Departamento de Matemática Aplicada, Universidad de Málaga, 29071 Málaga, Spain 0123456789().: V,-vol
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D. Girela, N. Merchán
If 0 < r < 1 and f ∈ Hol(D), we set 1/ p 2π 1 it p M p (r , f ) = | f (r e )| dt , 0 < p < ∞, 2π 0 M∞ (r , f ) = sup | f (z)|. |z|=r
If 0 < p ≤ ∞ the Hardy space H p consists of those f ∈ Hol(D) such that def
f H p = sup M p (r , f ) < ∞. 0 0 such that ρB ( f ) ≤ Aρ( f ), for all f ∈ X .
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Here, a linear functional L on X is said to be decent if it extends continuously to Hol(D). The space B M O A consists of those functions f in H 1 whose boundary values have bounded mean oscillation on the unit circle ∂D as defined by F. John and L. Nirenberg. There are many characterizations of B M O A functions. Let us mention the following: def
If f ∈ Hol(D), then f ∈ B M O A if and only if f B M O A = | f (0)| + ρ∗ ( f ) < ∞, where ρ∗ ( f ) = sup f ◦ ϕa − f (a) H 2 . a∈D
H∞
It is well known that ⊂ B M O A ⊂ B and that B M O A equipped with the seminorm ρ∗ is a Möbius invariant space. The space V M O A consists of those f ∈ B M O A such that lim|a|→1 f ◦ ϕa − f (a) H 2 = 0, it is the closure of the polynomials in the B M O A-norm. We mention [28] as a general reference for the space B M O A. Other important Möbius invariant spaces are the Besov spaces and the Q s spaces. For 1 < p < ∞, the analytic Besov space B p is defined as the set of all functions f p analy
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