Forward and Reverse Carleson Inequalities for Functions in Bergman Spaces and Their Derivatives

Luecking, Daniel H.

doi:10.2307/2374458

Cited by 179 publications

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“…A contradiction has been reached in that infinitely many disjoint D a/4 (ft,) have their centres in |z|<|z o |. The proof of the following lemma is quite similar to arguments used in [7] and [8]. (recall s is just the conjugate exponent of p/q).…”

Section: Theorem (E Amar) / / {Ft} Is a 5-separated Sequence Thenmentioning

confidence: 87%

Multipliers of Bergman spaces into Lebesgue spaces

Luecking

1986

Proceedings of the Edinburgh Mathematical Society

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Let U be the open unit disk in the complex plane endowed with normalized Lebesgue measure m. will denote the usual Lebesgue space with respect to m, with 0<p<+∞. The Bergman space consisting of the analytic functions in will be denoted . Let μ be some positivefinite Borel measure on U. It has been known for some time (see [6] and [9]) what conditions on μ are equivalent to the estimate: There is a constant C such thatprovided 0<p≦q.

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Section: Theorem (E Amar) / / {Ft} Is a 5-separated Sequence Thenmentioning

confidence: 87%

Multipliers of Bergman spaces into Lebesgue spaces

Luecking

1986

Proceedings of the Edinburgh Mathematical Society

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“…Finally, we need a special case of the Luecking's result [8] for p = q = 2 and α = 0 in which he characterized positive measures µ with the property…”

Section: Preliminariesmentioning

confidence: 99%

Generalized weighted composition operators on the Bergman space

Sharma

2011

Demonstratio Mathematica

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. In this paper, we characterize boundedness and compactness of D n ϕ,ψ on the Bergman space A 2 . We also compute the upper and lower bounds of essential norm of this operator on the Bergman space. IntroductionThroughout this paper, we denote by H(D), the space of holomorphic functions on the open unit disk D = {z ∈ C : |z| < 1} in the complex plane C. For z, w ∈ D, let β z (w) = (z − w)/(1 − zw) be the Möbius transformation of D which interchanges 0 and z. The nth derivative of β z is βπ r dr dθ be the normalized area measure on D. For 0 < p < ∞, let L p be the Lebesgue space which contains measurable functions f on D such thatthe weighted Bergman space with the norm defined asRecall that the Bergman space A 2 (D) = A 2 is a functional Hilbert space of analytic functions on D and K z (w) = 1/(1 − zw) 2 is the Bergman kernel. Moreover, See [6] and [23] for more details on Bergman spaces. Let ϕ and ψ be holomorphic maps on the open unit disk D such that ϕ(D) ⊂ D. For a non-negative integer n, we define a linear operator D n ϕ,ψ as follows:We call it generalized weighted composition operator, since it include many known operators. For example, if n = 0 and ψ ≡ 1, then we obtain the composition operator C ϕ induced by ϕ, defined asIf ψ = 1 and ϕ(z) = z, then D n ϕ,ψ = D n , the differentiation operator defined as D n f = f (n) . If n = 0, then we get the weighted composition operator ψC ϕ defined as, the product of multiplication and differentiation operator. We provide a unified way of treating these operators on the Bergman space A 2 . Composition and weighted composition operators have gained increasing attention during the last three decades, mainly due to the fact that they provide, just as, for example, Hankel and Toeplitz operators, ways and means to link classical function theory to functional analysis and operator theory.In fact, some of the well known conjectures can be linked to composition operators. Nordgren, Rosenthal and Wintrobe [11] have shown that the invariant subspace problem can be solved by classifying certain minimal invariant subspaces of certain composition operators on H 2 , whereas, Louis de Branges used composition operators to prove the Bieberbach conjecture. For general background on composition operators, we refer [2], [15] and references therein. Weighted composition operators also appear naturally in different contexts. For example, W. Smith in [20] showed that the Brennan's conjecture [1], which says that if G is a simply connected planar domain and g is a conformal map of G onto D, then Ì G |g ′ | p dA < ∞ holds for 4/3 < p < 4, is equivalent to the existence of self-maps of D that make certain weighted composition operators compact. The surjective isometries of Hardy and Bergman spaces are certain weighted composition operators (see

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“…It is well known (see [9], or Theorems 7.4 and 7.7 in [17] for example) that the Carleson measures for …”

Section: Preliminariesmentioning

confidence: 99%

Hankel Operators on Standard Bergman Spaces

Pau

2011

Complex Anal. Oper. Theory

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Abstract. We study Hankel operators on the standard Bergman spaces A 2 α , α > −1. A description of the boundedness and compactness of the (big) Hankel operator H f with general symbols f ∈ L 2 (D, dAα) is obtained. Also, we provide a new proof of a result of Arazy-FisherPeetre on the membership in Schatten p-classes of Hankel operators with conjugate analytic symbols.Mathematics Subject Classification (2010). 30H20, 47B10, 47B35.

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Forward and Reverse Carleson Inequalities for Functions in Bergman Spaces and Their Derivatives

Cited by 179 publications

References 11 publications

Multipliers of Bergman spaces into Lebesgue spaces

Multipliers of Bergman spaces into Lebesgue spaces

Generalized weighted composition operators on the Bergman space

Hankel Operators on Standard Bergman Spaces

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