Factorization of composition operators through Bloch type spaces

Jarchow, Hans; Riedl, Reinhard

doi:10.1215/ijm/1255986388

Cited by 14 publications

(10 citation statements)

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“…For 0 < a < 1, Jarchow and Riedl have given a characterization of those φ for which / i-> /' ο φ is a bounded operator from B a to a Hardy space H p , or equivalently Οφ : Ba+i -> H T is bounded. The case ρ = 2 can be stated as follows (see Theorem 1 and Corollary 4 of [9]):…”

Section: Zedmentioning

confidence: 99%

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COMPOSITION OPERATORS MAPPING INTO THE Qp SPACES

Smith¹,

Zhao²

1997

Analysis

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Composition operators are used to study the Qp spaces , which coincide with the Bloch space Β for ρ > 1 and are subspaces of BMOA for 0 < ρ < 1. Bounded composition operators from Bergman and Hardy spaces, and from Β to Qp are characterized by function theoretic properties of their inducing maps. Composition operators into weighted Dirichlet spaces and into little-oh subspaces of Qp, and compactness of these operators are also considered. The criteria for these operators to be bounded or compact are then interpreted geometrically when the symbol is univalent.AMS Classification: Primary 47B38 Secondary 30D55, 46E15 Smith-ZhaoHere dA(z) = dxdy/π is Lebesgue area measure normalized so that A(D) = 1. The subspace QPlo of Qp consists of those functions / such that the integral in the display above tends to 0 as |a| -+ 1. The Qp spaces can be thought of as Möbius invariant versions of weighted Dirichlet spaces, in the same way that the space of analytic functions of bounded mean oscillation, BMOA, is a Möbius invariant version of H 2 . For ρ > -1, the weighted Dirichlet space T>p is the Hilbert space of analytic functions on D satisfyingThe same space of functions results if (1 -|z|) is used in place of log(l/|z|) in the integral above. It is clear that this results in an equivalent norm, since these terms are comparable as |z| ->· 1 and the singularity of log(l/|z|) is integrable. In particular, T>o is the classical (unweighted) Dirichlet space, and it is well known that Τ>ι is the Hardy space H 2 and T>2 is the Bergman space A 2 . A change of variables shows that sup 11/ ο σα\\ 2 Vp = \\f\l 2 Qp.Thus the well known characterization (see [8]) of BMOA as those functions in H 2 whose norms after composition with conformai self-maps of the disk remain uniformly bounded is equivalent to the statement that Q\ = BMOA. Furthermore, it is known that when ρ > 1, Qp = β, the Bloch space of analytic functions satisfying ||/||B = sup(l -|z| 2 )|/'(z)| < oo, zÇD and QPl C QP2 if 0 < pi < P2 < 1; see [4] and [7]. Also, Qlfi = VMOA and for ρ > 1, Qpfi = Bo-Here VMOA is the subspace of BMOA consisting of functions of vanishing mean oscillation, and BQ is the "little Bloch" space of functions / analytic on D for which (1 -|z| 2 )|/'(z)| -> 0 as |z| -> 1. Finally, Qp is a Banach space with norm 11/11 = 1/(0)1 + ||/||q" and Qpfi is a closed subspace. Let ψ be an analytic self-map of D and let ϋφ be the induced composition operator. We are interested in the problem of using function theoretic properties of φ to determine when Οψ : <3P2 -> Qpi is bounded or compact, where 0 < p\ < P2-When p2 > 1 we have QP2 = B, and the problem is to characterize when Cv : Β -> Qp is bounded or compact. In this case, working with the Bloch norm, our methods provide results that are nearly complete. This extends recent work of K. Madigan and A. Matheson [12], who provided criteria equivalent to the compactness of Οφ : Β -¥ Β and Cv : Bo Bo. We also note that it is easy to see that Οφ is always bounded on Β and that Οφ is bounded on Bo if and only if φ € Bo...

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Section: Zedmentioning

confidence: 99%

“…(We note that there is a typographical error in the statement of Corollary 4 of [9]: H p and L p should read H& p and L^p, respectively.) Since 2?i = Η 2 , Theorem 1.3 with ρ = 1 can be viewed as the limiting case α -• 0 of Theorem 1.4.…”

Section: Theorem [9] S Let 0 < a < 1 And Let φ Be An Analytic Self-mamentioning

confidence: 99%

COMPOSITION OPERATORS MAPPING INTO THE Qp SPACES

Smith¹,

Zhao²

1997

Analysis

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“…In particular, in case is the identity self-map of D, Theorem 4.1 (i) requires that the derivative operator: f°f 0 be order-bounded from B to L p ðÞ, whereas Theorem 4.1 (ii) requires that be a ðB , pÞ-Carleson measure (see [19] for the definition). Accordingly, Theorem 4.1 extends [18,Corollary 4] and [19,Theorem 16] in very different ways. Indeed, if z 2 D, we have that j f 0 ðzÞj p is dominated by its average on each disc centered at z whose closure lies in D and, in particular, on such a disc of radius ð1 À jzjÞ=2.…”

Section: Embedding Via Derivationmentioning

confidence: 81%

“…(2) Theorem 4.1 characterizes actually either order-boundedness (for the definition, see [18]) or boundedness of the operator D : f°ð f Þ 0 , mapping B into L p ðÞ. In particular, in case is the identity self-map of D, Theorem 4.1 (i) requires that the derivative operator: f°f 0 be order-bounded from B to L p ðÞ, whereas Theorem 4.1 (ii) requires that be a ðB , pÞ-Carleson measure (see [19] for the definition).…”

Section: Embedding Via Derivationmentioning

confidence: 99%

BiBloch-type Maps: Existence and Beyond

Gauthier

Xiao

2002

Complex Variables, Theory and Application: An International Jou

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It is shown that for > 0 there are two constants c 1 , c 2 > 0 and a holomorphic map f from the unit disk D of C into C 2 such that c 1 ð1 À jzjÞ À j f 0 ðzÞj c 2 ð1 À jzjÞ À for all z 2 D. Moreover, this existence is effectively used in the study of invariance of the Bloch-type spaces under composition, but also in the discussion of embedding the Bloch-type spaces via derivation into the Lebesgue, mixed-norm and Coifman-Meyer-Stein tent spaces.

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“…Integrating this inequality with respect to t, applying Fubini's Theorem, Lemma 1 and Holder's inequality we get thaẗ 10, p. 437 and 1 y r z ; 1 y r z , we get that…”

Section: < < Z ª1mentioning

confidence: 95%

Composition Operators from Bloch Type Spaces to Hardy and Besov Spaces

Zhao

1999

Journal of Mathematical Analysis and Applications

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Boundedness and compactness of composition operators from Bloch type spaces to Hardy spaces and analytic Besov spaces are characterized by function theoretic properties of their inducing maps. For the case of the Bloch space, the characterizations involve the hyperbolic versions of Hardy and Besov classes. The conditions for the hyperbolic Besov classes are then interpreted geometrically when the symbols are univalent, and strict inclusion between different hyperbolic Besov classes is shown by an example. ᮊ

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Factorization of composition operators through Bloch type spaces

Cited by 14 publications

References 1 publication

COMPOSITION OPERATORS MAPPING INTO THE Qp SPACES

COMPOSITION OPERATORS MAPPING INTO THE Qp SPACES

BiBloch-type Maps: Existence and Beyond

Composition Operators from Bloch Type Spaces to Hardy and Besov Spaces

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