Mixed norm variable exponent Bergman space on the unit disc

2016 ◽  
Vol 61 (8) ◽  
pp. 1090-1106 ◽  
Author(s):  
A.N. Karapetyants ◽  
S.G. Samko
Keyword(s):  
Bergman Space ◽  
Unit Disc ◽  
Variable Exponent ◽  
Mixed Norm

scholarly journals Compact Operators on the Bergman Spaces with Variable Exponents on the Unit Disc of C

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Dieudonne Agbor
Keyword(s):  
Bergman Space ◽  
Unit Disc ◽  
Variable Exponent ◽  
Toeplitz Operators ◽  
Bergman Spaces ◽  
Berezin Transform ◽  
Variable Exponents ◽  
Bounded Operators ◽  
Finite Products ◽  
Finite Sum

We study the compactness of some classes of bounded operators on the Bergman space with variable exponent. We show that via extrapolation, some results on boundedness of the Toeplitz operators with general L1 symbols and compactness of bounded operators on the Bergman spaces with constant exponents can readily be extended to the variable exponent setting. In particular, if S is a finite sum of finite products of Toeplitz operators with symbols from class BT, then S is compact if and only if the Berezin transform of S vanishes on the boundary of the unit disc.

scholarly journals COMPOSITION OPERATORS BELONGING TO SCHATTEN CLASS 𝒮p

2010 ◽  
Vol 81 (3) ◽  
pp. 465-472
Author(s):  
CHENG YUAN ◽  
ZE-HUA ZHOU
Keyword(s):  
Bergman Space ◽  
Composition Operator ◽  
Unit Disc ◽  
Composition Operators ◽  
Schatten Class

AbstractWe investigate the composition operators Cφ acting on the Bergman space of the unit disc D, where φ is a holomorphic self-map of D. Some new conditions for Cφ to belong to the Schatten class 𝒮p are obtained. We also construct a compact composition operator which does not belong to any Schatten class.

scholarly journals Multipliers on Vector Valued Bergman Spaces

2002 ◽  
Vol 54 (6) ◽  
pp. 1165-1186 ◽  
Author(s):  
Oscar Blasco ◽  
José Luis Arregui
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Bergman Space ◽  
Unit Disc ◽  
Bergman Spaces ◽  
Complex Banach Space ◽  
Bounded Operators ◽  
Taylor Coefficients ◽  
Vector Valued

AbstractLet X be a complex Banach space and let Bp(X) denote the vector-valued Bergman space on the unit disc for 1 ≤ p < ∞. A sequence (Tn)n of bounded operators between two Banach spaces X and Y defines a multiplier between Bp(X) and Bq(Y) (resp. Bp(X) and lq(Y)) if for any function we have that belongs to Bq(Y) (resp. (Tn(xn))n ∈ lq(Y)). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces X and Y. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in Bp(X) are introduced.

On mixed norm Bergman–Orlicz–Morrey spaces

2018 ◽  
Vol 25 (2) ◽  
pp. 271-282 ◽  
Author(s):  
Alexey N. Karapetyants ◽  
Stefan G. Samko
Keyword(s):  
Unit Disc ◽  
Fourier Coefficients ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Bergman Projection ◽  
Mixed Norm ◽  
Taylor Coefficients ◽  
Type Spaces ◽  
Radial Variable

Abstract Following the ideas of our previous research, in this paper we continue the study of new Bergman-type spaces on the unit disc with mixed norm in terms of Fourier coefficients. Here we deal with the case where the sequence of norms of Fourier coefficients in the Orlicz–Morrey space in radial variable belongs to {l^{q}} . We study the boundedness of the Bergman projection and provide a description of functions in these spaces via the behavior of their Taylor coefficients.

scholarly journals ON THE COMPLEXITY OF FINDING A NECESSARY AND SUFFICIENT CONDITION FOR BLASCHKE-OSCILLATORY EQUATIONS

2014 ◽  
Vol 57 (3) ◽  
pp. 543-554
Author(s):  
JANNE HEITTOKANGAS ◽  
ATTE REIJONEN
Keyword(s):  
Differential Equation ◽  
Bergman Space ◽  
Nontrivial Solution ◽  
Unit Disc ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Number Of Zeros ◽  
Zero Sequence ◽  
Zeros Of Solutions ◽  
Necessary And Sufficient

AbstractIf A(z) belongs to the Bergman space , then the differential equation f″+A(z)f=0 is Blaschke-oscillatory, meaning that the zero sequence of every nontrivial solution satisfies the Blaschke condition. Conversely, if A(z) is analytic in the unit disc such that the differential equation is Blaschke-oscillatory, then A(z) almost belongs to . It is demonstrated that certain “nice” Blaschke sequences can be zero sequences of solutions in both cases when A ∈ or A ∉ . In addition, no condition regarding only the number of zeros of solutions is sufficient to guarantee that A ∈ .

scholarly journals Finite-rank intermediate Hankel operators on the Bergman space

2001 ◽  
Vol 25 (1) ◽  
pp. 19-31 ◽  
Author(s):  
Takahiko Nakazi ◽  
Tomoko Osawa
Keyword(s):  
Bergman Space ◽  
Orthogonal Projection ◽  
Finite Rank ◽  
Lebesgue Space ◽  
Unit Disc ◽  
Hankel Operator ◽  
Hankel Operators ◽  
Weighted Bergman Space ◽  
Open Unit ◽  
Open Unit Disc

LetL2=L2(D,r dr dθ/π)be the Lebesgue space on the open unit disc and letLa2=L2∩ℋol(D)be the Bergman space. LetPbe the orthogonal projection ofL2ontoLa2and letQbe the orthogonal projection ontoL¯a,02={g∈L2;g¯∈La2,   g(0)=0}. ThenI−P≥Q. The big Hankel operator and the small Hankel operator onLa2are defined as: forϕinL∞,Hϕbig(f)=(I−P)(ϕf)andHϕsmall(f)=Q(ϕf)(f∈La2). In this paper, the finite-rank intermediate Hankel operators betweenHϕbigandHϕsmallare studied. We are working on the more general space, that is, the weighted Bergman space.

scholarly journals Mixed norm Bergman–Morrey-type spaces on the unit disc

2016 ◽  
Vol 100 (1-2) ◽  
pp. 38-48 ◽  
Author(s):  
A. N. Karapetyants ◽  
S. G. Samko
Keyword(s):  
Unit Disc ◽  
Mixed Norm ◽  
Type Spaces

scholarly journals Linear functionals on some weighted Bergman spaces

1990 ◽  
Vol 42 (3) ◽  
pp. 417-425 ◽  
Author(s):  
Maher M.H. Marzuq
Keyword(s):  
Bergman Space ◽  
Analytic Functions ◽  
Unit Disc ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Linear Functionals

The weighted Bergman space Ap, α, 0 < p < 1, a > −1 of analytic functions on the unit disc Δ in C is an F-space. We determine the dual of Ap, α explicitly.

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