Norm of the Bergman Projection

This paper deals with the the norm of the weighted Bergman projection operator $P_{\alpha}:L^\infty(B)\rightarrow\mathscr{B}$ where $\alpha > - 1$ and $\mathscr{B}$ is the Bloch space of the unit ball $B$ of the $\mathsf{C}^n$. We consider two Bloch norms, the standard Bloch norm and invariant norm w.r.t. automorphisms of the unit ball. Our work contains as a special case the main result of the recent paper [6].

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A Sharp Constant for the Bergman Projection

Canadian Mathematical Bulletin ◽

10.4153/cmb-2014-034-0 ◽

2015 ◽

Vol 58 (1) ◽

pp. 128-133 ◽

Cited By ~ 1

Author(s):

Marijan Marković

Keyword(s):

Unit Ball ◽

Besov Space ◽

Projection Operator ◽

Bergman Projection ◽

Hyperbolic Metric ◽

Sharp Constant ◽

The Hyperbolic Metric

AbstractFor the Bergman projection operator P we prove thatHere λ stands for the hyperbolic metric in the unit ball B of Cn, and B1 denotes the Besov space with an adequate semi-norm. We also consider a generalization of this result. This generalizes some recent results due to Perälä.

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Besov-type characterisations for the Bloch space

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700003324 ◽

1989 ◽

Vol 39 (3) ◽

pp. 405-420 ◽

Cited By ~ 45

Author(s):

Karel Stroethoff

Keyword(s):

Unit Ball ◽

Besov Space ◽

Analytic Functions ◽

Unit Disc ◽

Bloch Space ◽

Bloch Functions ◽

Little Bloch Space ◽

Special Case

We will prove local and global Besov-type characterisations for the Bloch space and the little Bloch space. As a special case we obtain that the Bloch space consists of those analytic functions on the unit disc whose restrictions to pseudo-hyperbolic discs (of fixed pseudo-hyperbolic radius) uniformly belong to the Besov space. We also generalise the results to Bloch functions and little Bloch functions on the unit ball in . Finally we discuss the related spaces BMOA and VMOA.

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A Theorem of Nehari Type on Weighted Bergman Spaces of the Unit Ball

Abstract and Applied Analysis ◽

10.1155/2008/538573 ◽

2008 ◽

Vol 2008 ◽

pp. 1-7

Author(s):

Yufeng Lu ◽

Jun Yang

Keyword(s):

Linear Operator ◽

Unit Ball ◽

Bounded Linear Operator ◽

Hankel Operator ◽

Bergman Spaces ◽

Bergman Projection ◽

Weighted Bergman Spaces ◽

Bounded Linear ◽

Weighted Bergman Projection

This paper shows that ifSis a bounded linear operator acting on the weighted Bergman spacesAα2on the unit ball inℂnsuch thatSTzi=Tz¯iS (i=1,…,n), whereTzi=zifandTz¯i=P(z¯if); and wherePis the weighted Bergman projection, thenSmust be a Hankel operator.

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Norm inequalities for positive semi-definite matrices and a question of Bourin II

International Journal of Mathematics ◽

10.1142/s0129167x21500439 ◽

2021 ◽

pp. 2150043

Author(s):

Mostafa Hayajneh ◽

Saja Hayajneh ◽

Fuad Kittaneh

Keyword(s):

Affirmative Answer ◽

Unitarily Invariant Norm ◽

Norm Inequalities ◽

Invariant Norm ◽

Special Case

Let [Formula: see text] and [Formula: see text] be [Formula: see text] positive semi-definite matrices. It is shown that [Formula: see text] for every unitarily invariant norm. This gives an affirmative answer to a question of Bourin in a special case. It is also shown that [Formula: see text] for [Formula: see text] and for every unitarily invariant norm.

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On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2008.12.059 ◽

2009 ◽

Vol 354 (2) ◽

pp. 426-434 ◽

Cited By ~ 87

Author(s):

Stevo Stević

Keyword(s):

Unit Ball ◽

Bloch Space ◽

Type Operator ◽

Integral Type ◽

Type Spaces

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On the p-norm of an Integral Operator in the Half Plane

Canadian Mathematical Bulletin ◽

10.4153/cmb-2011-186-3 ◽

2013 ◽

Vol 56 (3) ◽

pp. 593-601 ◽

Cited By ~ 3

Author(s):

Congwen Liu ◽

Lifang Zhou

Keyword(s):

Integral Operator ◽

Half Plane ◽

Integral Operators ◽

Bergman Projection ◽

Partial Answer ◽

Upper Half Plane ◽

Weighted Bergman Projection

Abstract.We give a partial answer to a conjecture of Dostanić on the determination of the norm of a class of integral operators induced by the weighted Bergman projection in the upper half plane.

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Holomorphic Functions with Positive Real Part

Canadian Journal of Mathematics ◽

10.4153/cjm-1982-001-1 ◽

1982 ◽

Vol 34 (1) ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Eric Sawyer

Keyword(s):

Unit Ball ◽

Hardy Spaces ◽

Convex Cone ◽

Holomorphic Functions ◽

Positive Real Part ◽

Positive Real ◽

Spaces Of Holomorphic Functions ◽

Extreme Element ◽

Image Position ◽

Special Case

The main purpose of this note is to prove a special case of the following conjecture.Conjecture. If F is holomorphic on the unit ball Bn in Cn and has positive real part, then F is in Hp(Bn) for 0 < p < ½(n + 1).Here Hp(Bn) (0 < p < ∞) denote the usual Hardy spaces of holomorphic functions on Bn. See below for definitions. We remark that the conjecture is known for 0 < p < 1 and that some evidence for it already exists in the literature; for example [1, Theorems 3.11 and 3.15] where it is shown that a particular extreme element of the convex cone of functionsis in Hp(B2) for 0 < p < 3/2.

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