# Superposition Operators, Hardy Spaces, And Dirichlet Type Spaces

Petros Galanopoulos, Daniel Girela, Maria Auxiliadora Marquez

Published 2018 · Mathematics

Abstract For 0 p ∞ and α > − 1 the space of Dirichlet type D α p consists of those functions f which are analytic in the unit disc D and satisfy ∫ D ( 1 − | z | ) α | f ′ ( z ) | p d A ( z ) ∞ . The space D p − 1 p is the closest one to the Hardy space H p among all the D α p . Our main object in this paper is studying similarities and differences between the spaces H p and D p − 1 p ( 0 p ∞ ) regarding superposition operators. Namely, for 0 p ∞ and 0 s ∞ , we characterize the entire functions φ such that the superposition operator S φ with symbol φ maps the conformally invariant space Q s into the space D p − 1 p , and, also, those which map D p − 1 p into Q s and we compare these results with the corresponding ones with H p in the place of D p − 1 p . We also study the more general question of characterizing the superposition operators mapping D α p into Q s and Q s into D α p , for any admissible triplet of numbers ( p , α , s ) .

This paper references

The nonlinear superposition operator acting on Bergman spaces

Gianni Camera (1994)

Geometric Q functions

J. Xiao (2006)

10.1007/BF02355826

Multiplication and division in the space of analytic functions with area integrable derivative, and in some related spaces

S. A. Vinogradov (1997)

10.1515/9783111576855-015

J

Seguin Hen (1824)

10.1017/S0305004199004338

On the zeros of Bloch functions

Maria Nowak (2000)

10.1006/jfan.1997.3114

Bounded Functions in Mo bius Invariant Dirichlet Spaces

Artur Nicolau (1997)

10.1515/crll.1974.270.12

On Bloch functions and normal functions.

Christian Pommerenke (1974)

10.1216/rmjm/1181072070

Some Subclasses of BMOA and their Characterization in Terms of Carleson Measures

Rauno Aulaskari (1996)

10.1006/jfan.1999.3490

Carleson Measures and Multipliers for Dirichlet Spaces

Zhijian Wu (1999)

10.2140/pjm.2000.194.491

The Qp corona theorem

Jie Xiao (2000)

10.1007/978-1-4612-0497-8

Theory of Bergman Spaces

Haakan Per Jan Hedenmalm (2000)

10.1090/surv/138

Operator theory in function spaces

Kehe Zhu (1990)

zeros of normal functions, Ann

M. Nowak (2002)

On Qp spaces and pseudoanalytic extension

Konstantin M. Dyakonov (2000)

Some results on Q(p) spaces, 0lpl1

Matts R. Essén (1997)

On zeros of normal functions

Maria Nowak (2002)

10.1080/17476939208814602

Growth of the derivative of bounded analytic functions

Daniel Girela (1992)

10.1007/978-1-4612-0497-8_1

The Bergman Spaces

Haakan Per Jan Hedenmalm (2000)

10.1112/s0024609301231002

THEORY OF BERGMAN SPACES (Graduate Texts in Mathematics 199) By HAAKAN HEDENMALM, BORIS KORENBLUM and KEHE ZHU: 286 pp., £37.50, ISBN 0-387-98791-6 (Springer, New York, 2000).

Peter Duren (2002)

10.1007/S00605-012-0441-6

Superposition operators between weighted Banach spaces of analytic functions of controlled growth

Jose Bonet (2013)

Some results on Qp-spaces

Messe Essen (1997)

10.1112/blms/10.2.219

On Analytic Functions of Bounded Mean Oscillation

Walter Kurt Hayman (1978)

10.1007/b87877

Holomorphic Q Classes

Jie Xiao (2001)

10.1007/S11118-008-9081-9

Univalent Interpolation in Besov Spaces and Superposition into Bergman Spaces

Stephen M. Buckley (2008)

Nonlinear superposition on spaces of analytic functions

G. A. Cámera (1994)

10.1016/J.JMAA.2014.03.058

Operator theoretic differences between Hardy and Dirichlet-type spaces

Jos'e 'Angel Pel'aez (2013)

10.1524/anly.1995.15.2.101

ON SUBSPACES AND SUBSETS OF BMOA AND UBC

Rauno Aulaskari (1995)

10.1016/0022-247X(72)90081-9

The dual of an inequality of Hardy and Littlewood and some related inequalities

Thomas Muirhead Flett (1972)

10.1016/s0079-8169(08)x6157-4

Theory of Hp Spaces

Peter L. Duren (2000)

10.1007/s00020-005-1391-3

Carleson Measures for Spaces of Dirichlet Type

Daniel Girela (2006)

10.1515/crll.2002.102

On univalent functions in some Möbius invariant spaces

Juan Jesús Donaire (2002)

10.1201/9781420034875.ch9

On univalent functions

N. Ayırtman (1965)

Superposition operators on Dirichlet type spaces

Stephen M. Buckley (2001)

10.1017/S1446788700014105

Growth properties and sequences of zeros of analytic functions in spaces of Dirichlet type

Daniel Girela Alvarez (2006)

10.1017/s030500419800334x

FRACTIONAL INTEGRATION, DIFFERENTIATION, AND WEIGHTED BERGMAN SPACES

Stephen M. Buckley (1999)

10.1007/BF02385476

Superposition operators between the Bloch space and Bergman spaces

Venancio Alvarez (2004)

10.1162/003465399558490

What is Fractional Integration?

William R. Parke (1999)

10.5186/aasfm.2014.3912

GENERALIZED HILBERT OPERATORS

Petros Galanopoulos (2014)

10.1215/ijm/1258131055

Univalent functions, Hardy spaces and spaces of Dirichlet type

Albert Baernstein (2004)

10.1007/S00209-008-0338-1

Multipliers of Möbius invariant Qs spaces

Jordi Pau (2008)

10.1112/plms/s2-42.1.52

Theorems on Fourier Series and Power Series (II)

John Edensor Littlewood (1937)

10.1006/jmaa.2000.7259

Taylor Coefficients and Mean Growth of the Derivative of Qp Functions

Rauno Aulaskari (2001)

Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal

Rauno Aulaskari (1994)

Vukotić, Superposition operators on Dirichlet type spaces. In: Papers on Analysis: a volume dedicated to Olli Martio on the occasion of his 60th Birthday, 41–61

S. M. Buckley (2001)

10.1016/J.JMAA.2009.10.011

Superposition operators between Qp spaces and Hardy spaces

Daniel Girela (2010)

This paper is referenced by