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The Resource Taylor Coefficients and Coefficient Multipliers of Hardy and Bergman-Type Spaces

Taylor Coefficients and Coefficient Multipliers of Hardy and Bergman-Type Spaces

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Taylor Coefficients and Coefficient Multipliers of Hardy and Bergman-Type Spaces
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Taylor Coefficients and Coefficient Multipliers of Hardy and Bergman-Type Spaces
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eng
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MiAaPQ
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dictionaries
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RSME Springer Series
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v.2
Taylor Coefficients and Coefficient Multipliers of Hardy and Bergman-Type Spaces
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Taylor Coefficients and Coefficient Multipliers of Hardy and Bergman-Type Spaces
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http://libproxy.rpi.edu/login?url=https://ebookcentral.proquest.com/lib/rpi/detail.action?docID=4773794
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txt
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rdacontent
Contents
  • Preface -- Acknowledgments -- Contents -- Introduction -- References{u0082}1. Hardy, G.H., Littlewood, J.E.: Some properties of fractional integrals, II. Math. Z. 34, 403-439 (1932){u0082}2. Littlewood, J.E., Paley, R.E.A.C.: Theorems on Fourier series and power series, II. Proc. London Math. Soc. 42, 52-89 (1936){u0082}3. Duren, P.L.: Theory of Hp Spaces. Academic Press, New York (1970) -- reprinted with supplement by Dover Publications, Mineola, NY, (2000){u0082}4. Garnett, J.B.: Bounded Analytic Functions. Academic Press, New York (1981){u0082}5. Koosis, P.: Introduction to Hp Spa -- 1 Basic Spaces. Multipliers -- 1.1 A Family of Normed and Mixed-Norm Spaces -- 1.2 Quasi-Banach and p-Banach Spaces. F-spaces -- 1.3 Coefficient Multipliers -- 1.3.1 Definitions and First Properties -- 1.3.2 Three Interpretations of Coefficient Multipliers -- 1.3.3 Duality and Multipliers -- 1.3.4 A Remark on Duals -- References -- 2 The Poisson Integral -- 2.1 The Poisson Integral of a Continuous Function -- 2.2 Borel Measures and the Space h1 -- 2.2.1 The Poisson Integral of a Measure -- 2.2.2 The Banach-Alaoglu Theorem -- 2.2.3 The Riesz-Herglotz Theorem -- 2.2.4 The Poisson-Stieltjes Integral -- 2.3 The Poisson Integral, Lp(mathbbT), and hp Spaces When 1<pleinfty -- 2.3.1 The Poisson Integral of a Function -- 2.3.2 Weak- Convergence Properties of the Poisson Integral -- 2.4 The Maximal Function of a Measure on mathbbT -- 2.4.1 The Maximal Function of a Measure on mathbbT -- 2.4.2 Lorentz Spaces -- 2.4.3 The Maximal Theorem -- 2.5 Nontangential Maximal Function and Fatou's Theorem -- 2.5.1 Nontangential Maximal Function -- 2.5.2 Nontangential Limits -- 2.5.3 The Space hpmax. Atomic Decomposition -- 2.6 Some Useful Practical Facts -- 2.7 Historical and Bibliographical Notes -- 2.8 Exercises -- References -- 3 Subharmonic and h-Subharmonic Functions
  • 3.1 The Pseudohyperbolic and Hyperbolic Metric -- 3.1.1 The Pseudohyperbolic Metric -- 3.1.2 The Hyperbolic Metric -- 3.2 Behavior of Subharmonic Functions -- 3.2.1 Harmonic and h-Harmonic Functions -- 3.2.2 Subharmonic Functions -- 3.2.3 Integral Means of Subharmonic Functions -- 3.2.4 Smooth Subharmonic Functions -- 3.3 h-Subharmonic Functions -- 3.4 Subharmonic Behavior -- 3.5 The Harmonic Hardy Spaces hp -- 3.6 Maximal Theorems -- 3.7 Identities of Littlewood-Paley and Hardy-Stein -- 3.7.1 Littlewood-Paley Identity -- 3.7.2 Hardy-Stein Identities -- 3.7.3 The Hardy Space Connection -- 3.8 Some Useful Practical Facts -- 3.9 Historical and Bibliographical Notes -- 3.10 Exercises -- References -- 4 Hardy Spaces of Analytic Functions -- 4.1 Hardy Spaces -- 4.1.1 Fatou's Theorem for Hardy Spaces -- 4.1.2 Inner and Outer Functions -- 4.1.3 The Nevanlinna Class -- 4.2 The Riesz Projection Theorem -- 4.2.1 Harmonic Conjugates -- 4.2.2 The Riesz Projection Operator -- 4.2.3 The Cauchy-Szegö Transform -- 4.2.4 The Hilbert Operator -- 4.2.5 Norm Estimate for the Riesz Projection Operator -- 4.2.6 The Dual Space of Hp, 1<p<infty -- 4.2.7 A Theorem of Littlewood-Paley Type on Dyadic Blocks -- 4.3 On Harmonic Conjugates -- 4.3.1 Kolmogorov's Theorem on the Growth of Harmonic Conjugates -- 4.3.2 On Harmonic Conjugates with Exponential Mean Growth -- 4.4 Growth of Integral Means of Hp Functions -- 4.4.1 Growth of Integral Means of Hp Functions -- 4.4.2 A Theorem of Hardy and Littlewood -- 4.4.3 A Theorem of Littlewood and Paley -- 4.4.4 An Inequality for the Integral Means of the Hadamard Product of Two Power Series -- 4.4.5 On Equivalence of Different Forms of Fractional Derivatives -- 4.5 Some Useful Practical Facts -- 4.6 Historical and Bibliographical Notes -- 4.7 Exercises -- References -- 5 Carleson Measures, Mean Oscillation Spaces and Duality
  • 5.1 Carleson Measures for Hardy Spaces -- 5.1.1 Carleson Measures for Hardy Spaces -- 5.1.2 Vanishing Carleson Measures -- 5.2 The Space BMO -- 5.2.1 The Space BMO(mathbbT) -- 5.2.2 The Conformally Invariant BMO Space -- 5.2.3 The Analytic BMO Space -- 5.2.4 BMOA as a Dual of H1 -- 5.3 Functions of Vanishing Mean Oscillation -- 5.3.1 Functions of Vanishing Mean Oscillation -- 5.3.2 The Space VMOA -- 5.3.3 H1 as a Dual of VMOA -- 5.4 Historical and Bibliographical Notes -- 5.5 Exercises -- References -- 6 Polynomial Approximation and Taylor Coefficients of Hp Functions -- 6.1 Approximation of Hp Functions by Polynomials -- 6.1.1 Approximation of Hp, 1 < p < infty, Functions by Polynomials -- 6.1.2 Approximation of Hp, 0 < p leq1, Functions by Polynomials -- 6.1.3 S̀̀mooth'' Cesàro Means -- 6.2 Taylor Coefficients of Hp Functions -- 6.2.1 Hausdorff--Young Theorem -- 6.2.2 Lacunary Series in Hp -- 6.2.3 Taylor Coefficients of Hp Functions: Necessary Conditions -- 6.2.4 Taylor Coefficients of Hp Functions: Sufficient Conditions -- 6.2.5 Monotonic Sequences in Hp -- 6.2.6 Characterizations of the Coefficients of Bounded Analytic Functions and Inner Functions -- 6.3 Solid Hulls and Cores of Hardy Spaces -- 6.3.1 Solid Hulls of Hardy Spaces -- 6.3.2 Solid Cores of Hardy Spaces -- 6.4 Taylor Coefficients of BMOA Functions -- 6.4.1 The Solid Core of the Space BMOA -- 6.5 Historical and Bibliographical Notes -- 6.6 Exercises -- References -- 7 The Mixed Norm Spaces Hp,q,&#xdc; -- 7.1 Definitions and First Properties -- 7.2 Bergman Spaces -- 7.3 A Projection Theorem for Hp,q,&#xdc; -- 7.4 Decomposition Theorems for Hp,q,&#xdc; -- 7.5 Functions with a Fractional Derivative in Mixed Norm Spaces -- 7.5.1 The Bloch Space -- 7.5.2 Lipschitz Spaces -- 7.5.3 Besov Spaces -- 7.5.4 A Remark on Terminology -- 7.6 Some Useful Practical Facts
  • 7.6.1 An Inclusion Between Hardy and Bergman Spaces and the Isoperimetric Inequality -- 7.6.2 The Bloch Space, Taylor Coefficients, and Lacunary Series -- 7.7 Historical and Bibliographical Notes -- 7.8 Exercises -- References -- 8 Hp,q,&#xdc; as a Sequence Space -- 8.1 Some Special Sequences Defining Hp,q,&#xdc; Functions -- 8.1.1 Lacunary Series in Hp,q,&#xdc; Spaces -- 8.1.2 Monotonic Sequences in Hp,q,&#xdc; -- 8.2 The Solid Hull of Hp,q,&#xdc; -- 8.2.1 Solid Hulls of the Hardy--Lorentz Spaces Hp,q -- 8.3 Solid Cores of Hp,q,&#xdc; Spaces -- 8.4 Taylor Coefficients of Functions in Bergman Spaces -- 8.5 Historical and Bibliographical Notes -- 8.6 Exercises -- References -- 9 Tensor Products and Multipliers -- 9.1 Tensor Product of Two Spaces -- 9.2 Historical and Bibliographical Notes -- 9.3 Exercises -- References -- 10 Duality and Multipliers -- 10.1 Duality for Hardy Spaces and Multipliers of Hardy Spaces into mathcalA and Hinfty -- 10.2 Duality for Mixed Norm Spaces -- 10.2.1 Dual Spaces for Mixed Norm Spaces -- 10.2.2 The Dual Space of Hp,infty,&#xdc;0, 1leqp leqinfty -- 10.3 Multipliers (Hp,q,&#xdc;,Hinfty) -- 10.4 Duality and Multipliers for Hardy and Mixed Norm Spaces: Summary -- 10.4.1 Duality and Multipliers for Mixed Norm Spaces -- 10.4.2 Duality and Multipliers for Hardy--Lorentz Spaces -- 10.5 Historical and Bibliographical Notes -- 10.6 Exercises -- References -- 11 Multipliers from Hp and Hp,q,&#xdc; Spaces to ells -- 11.1 Multipliers into Sequence Spaces -- 11.2 Multipliers from Hp to ellu -- 11.3 Multipliers from Hp,q,&#xdc; to ellu -- 11.4 Historical and Bibliographical Notes -- 11.5 Exercises -- References -- 12 Multiplier Spaces (Hp,q,&#xdc;,Hu,v,&#xdd;) and (Hp,Hu) -- 12.1 Fractional Integrals of Hp Functions -- 12.2 General Theorems -- 12.3 A Characterization of Multipliers (Hp,q,&#xdc;,Hu,v,&#xdd;) and (Hp,Hu), 0<p leq1 -- 12.3.1 Multipliers (Hp,q,&#xdc;,Hu,v,&#xdd;) Case 0<pleq1
  • 12.3.2 Multipliers (Hp,Hu,v,&#xdd;): The Case 0<pleq1 -- 12.3.3 Multipliers (Hp,Hu), 0<p<1 -- 12.3.4 Multipliers of the Hardy Space H1 -- 12.3.5 Multipliers (Hp,q,&#xdc;,Hu): The Case 0<pleq1 -- 12.4 Multipliers (Hp,q,&#xdc;,Hu,v,&#xdd;) and (Hp,Hu) When 2leqp leqinfty -- 12.4.1 Multipliers (Hp,q,&#xdc;,Hu,v,&#xdd;), 2leqp leqinfty -- 12.4.2 Multipliers of the Hardy Space Hp, 2leqpleqinfty -- 12.4.3 Multipliers (Hp,q,&#xdc;,Hu), 2leqpleqinfty -- 12.4.4 Multipliers (Hp,Hu,v,&#xdd;), 2leqpleqinfty -- 12.5 More on Multipliers into Hinfty,v, &#xdd; -- 12.5.1 Multipliers (Hp,q,&#xdc;,Hinfty,v,&#xdd;), 1<pleqinfty -- 12.5.2 Multipliers (Hp,Hinfty,v,&#xdd;), 1< pleqinfty -- 12.6 Remarks on the Multipliers of Hardy and Bergman Spaces -- 12.6.1 Solidity of (Hp,Hq) Spaces -- 12.6.2 More on Fractional Integration on Hp Spaces -- 12.6.3 Remarks on (Ap,Aq) Multipliers -- 12.7 Self-multipliers of Bergman Spaces -- 12.7.1 Isometric Coefficient Multipliers of Weighted Bergman Spaces -- 12.7.2 Sequences of Bounded Variation as (Non-)multipliers -- 12.8 Historical and Bibliographical Notes -- 12.9 Exercises -- References -- 13 Multipliers of Some Large Spaces of Analytic Functions -- 13.1 Multipliers in the Space of Analytic Functions with Exponential Mean Growth -- 13.2 Historical and Bibliographical Notes -- 13.3 Exercises -- References -- 14 The Hilbert Matrix Operator -- 14.1 Hankel Operators on ell2 (or H2) and Nehari's Theorem -- 14.2 The Hilbert Matrix as an Operator on ellp Spaces -- 14.3 The Hilbert Matrix as an Operator on Hardy Spaces -- 14.3.1 Boundedness of Hankel Operators on Hardy Spaces -- 14.3.2 The Norm of the Hilbert Matrix on Hardy Spaces -- 14.3.3 An Application to Hardy's Inequality -- 14.4 Hilbert Matrix as an Operator on Bergman Spaces -- 14.4.1 Boundedness and Norm on Bergman Spaces -- 14.5 Historical and Bibliographical Notes -- 14.6 Exercises -- References -- Additional References -- Index
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