Theory of bergman spaces, haakan hedenmalm, boris korenblum, kehe zhu

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WHITEHEAD Elements of Homotopy (continued after index) Haakan Hedenmalm Boris Korenblum Kehe Zhu Theory of Bergman Spaces With Illustrations Springer Haakan Hedenmalm Department of Mathematics Lund University Lund, S-22100 Sweden Boris Korenblum Kehe Zhu Department of Mathematics State University of New York at Albany Albany, NY 12222-0001 USA Editorial Board S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA F.W Gehring Mathematics Department Bast Hall University of Michigan Ann Arbor, MI 48109 USA K.A Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject Classification (2000): 47-01, 47A15, 32A30 Library of Congress Cataloging-in-Publication Data Hedenmalm, Haakan Theory of Bergman spaces I Haakan Hedenmalrn, Boris Korenblurn, Kehe Zhu p cm - (Graduate texts in rnathernatics ; 199) Includes bibliographical references and index ISBN 978-1-4612-6789-8 ISBN 978-1-4612-0497-8 (eBook) DOI 10.1007/978-1-4612-0497-8 Bergman kernel functions IV Series QA33l H36 2000 5l5-dc2l I Korenblurn, Boris 11 Zhu, Kehe, 1961- III Title 99-053568 Printed on acid-free paper © 2000 Springer Science+Business Media New York Originally published by Springer-Verlag New York Berlin Heidelberg in 2000 Softcover reprint of the hardcover 1st edition 2000 All rights reserved This work rnay not be translated or copied in whole or in part without the written permission of the Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar rnethodology now known or hereafter developed is forbidden The use of general descriptive narnes, trade narnes, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Production rnanaged by Jenny Wolkowicki; rnanufacturing supervised by Jeffrey Taub Photocornposed copy prepared frorn the authors' LaTeX files 543 ISBN 978-1-4612-6789-8 Preface Their memorials are covered by sand, their rooms are forgotten But their names live on by the books they wrote, for they are beautiful (Egyptian poem, 1500 1000 BC) The theory of Bergman spaces experienced three main phases of development during the last three decades The early 1970's marked the beginning of function theoretic studies in these spaces Substantial progress was made by Horowitz and Korenblum, among others, in the areas of zero sets, cyclic vectors, and invariant subspaces An influential presentation of the situation up to the mid 1970 's was Shields' survey paper "Weighted shift operators and analytic function theory" The 1980's saw the thriving of operator theoretic studies related to Bergman spaces The contributors in this period are numerous; their achievements were presented in Zhu's 1990 book "Operator Theory in Function Spaces" The research on Bergman spaces in the 1990 's resulted in several breakthroughs, both function theoretic and operator theoretic The most notable results in this period include Seip's geometric characterization of sequences of interpolation and sampling, Hedenmalm's discovery of the contractive zero divisors, the relationship between Bergman-inner functions and the biharmonic Green function found by vi Preface Duren, Khavinson, Shapiro, and Sundberg, and deep results concerning invariant subspaces by Aleman, Borichev, Hedenmalm, Richter, Shimorin, and Sundberg Our purpose is to present the latest developments, mostly achieved in the 1990's, in book form In particular, graduate students and new researchers in the field will have access to the theory from an almost self-contained and readable source Given that much of the theory developed in the book is fresh, the reader is advised that some of the material covered by the book has not yet assumed a final form The prerequisites for the book are elementary real, complex, and functional analysis We also assume the reader is somewhat familiar with the theory of Hardy spaces, as can be found in Duren's book "Theory of HP Spaces", Garnett's book "Bounded Analytic Functions", or Koosis' book "Introduction to if Spaces" Exercises are provided at the end of each chapter Some of these problems are elementary and can be used as homework assignments for graduate students But many of them are nontrivial and should be considered supplemental to the main text; in this case, we have tried to locate a reference for the reader We thank Alexandru Aleman, Alexander Borichev, Bernard Pinchuk, Kristian Seip, and Sergei Shimorin for their help during the preparation of the book We also thank Anders Dahlner for assistance with the computer generation of three pictures, and Sergei Treil for assistance with one January 2000 Haakan Hedenmalm Boris Korenblum Kehe Zhu Contents Preface v The Bergman Spaces Bergman Spaces 1.1 1.2 1.3 1.4 1.5 1.6 Some LP Estimates The Bloch Space Duality of Bergman Spaces Notes Exercises and Further Results The Berezin Transform 2.1 Algebraic Properties 2.2 Harmonic Functions 2.3 Carleson-Type Measures 2.4 BMO in the Bergman Metric 2.5 A Lipschitz Estimate 2.6 Notes 2.7 Exercises and Further Results A P -Inner Functions A~-Inner Functions An Extremal Problem The Biharmonic Green Function The Expansive Multiplier Property 3.1 3.2 3.3 3.4 13 17 22 23 28 28 32 38 42 46 49 49 52 52 55 59 66 Contents viii 3.5 3.6 3.7 3.8 3.9 Contractive Zero Divisors in A P An Inner-Outer Factorization Theorem for AP Approximation of Subinner Functions Notes Exercises and Further Results Zero Sets 4.1 Some Consequences of Jensen's Formula 4.2 Notions of Density The Growth Spaces A -Ci and A -00 • • 4.3 4.4 A -Ci Zero Sets, Necessary Conditions A -Ci Zero Sets, a Sufficient Condition 4.5 4.6 Zero Sets for Ag 4.7 The Bergman-Nevanlinna Class 4.8 Notes 4.9 Exercises and Further Results 71 78 86 94 95 98 98 104 110 112 119 128 131 133 134 Interpolation and Sampling Interpolation Sequences for A-Ci Sampling Sets for A-Ci Interpolation and Sampling in Ag Hyperbolic Lattices Notes Exercises and Further Results 136 136 152 155 165 171 172 Invariant Subspaces Invariant Subspaces of Higher Index 6.1 6.2 Inner Spaces in A~ A Beurling-Type Theorem 6.3 6.4 Notes Exercises and Further Results 6.5 176 176 180 181 186 187 5.1 5.2 5.3 5.4 5.5 5.6 7.1 7.2 7.3 7.4 7.5 7.6 Cyclicity Cyclic Vectors as Outer functions Cyclicity in A P Versus in A- oo Premeasures for Functions in A- oo Cyclicity in A- oo Notes Exercises and Further Results 190 190 191 193 208 214 214 Invertible Noncyclic Functions An Estimate for Harmonic Functions 8.1 8.2 The Building Blocks 8.3 The Basic Iteration Scheme 8.4 The Mushroom Forest 216 217 219 222 230 Contents 8.5 8.6 8.7 8.8 Finishing the Construction Two Applications Notes Exercises and Further Results ix 235 238 239 240 242 Logarithmically Subbarmonic Weights 9.1 Reproducing Kernels 9.2 Green Functions with Smooth Weights 9.3 Green Functions with General Weights 9.4 An Application 9.5 Notes 9.6 Exercises and Further Results 242 253 262 267 269 269 Rderences 274 Index 282 References 275 [13] S Axler, The Bergman space, the Bloch space, and commutators of multiplication operators, Duke Math J 53 (1986), 315-332 [14] S Axler, Bergman spaces and their operators, Surveys of Some Recent Results in Operator Theory, Volume 1, (J.B Conway and B.B Morrel, editors), Pitman Research Notes in Math 171, 1988, 1-50 [15] D Bekolle, C A Berger, L A Coburn, and K Zhu, BMO in the Bergman metric on bounded symmetric domains, J Funct Anal 93 (1990), 310-350 [16] H Bercovici, C Foia~, and C Pearcy, Dual algebras with applications to invariant subspaces and dilation theory, CBMS Regional Conference Series in Mathematics, 56; published for the Conference Board of the 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19-24 [143] K Zhu, Maximal inner spaces and Hankel operators on the Bergman space, Integral Equations Operator Theory 31 (1998), 371-387 [144] K Zhu, A sharp estimate for extremal functions, Proc Amer Math Soc., in print Index A2 -subinner function, 52 AP-inner function, 52 Ag+, limit of Bergman spaces, 130 Ag-, limit of Bergman spaces, 130 B2(w), large weighted Bergman space, 245 B p, the Besov space, 23 C k (X),60 C a (r),137 D(z, r), hyperbolic disk, 38 D+(A), upper asymptotic K-density, o 105 D;(r), upper Seip density, 147 D;t(r), uniform separating upper asymptotic K-density of f, 140 D-(A), lower asymptotic K-density, 105 D-; (f), lower Seip density, 153 Da, "fractional differentiation", 18 Da, "fractional integration", 19 G(z, w), Green function for the Laplacian, 60 G[f], Green potential, 60 G A, zero divisor for A, 57 G /, the extremal function for I, 57 H P, the Hardy space, 54 BeAz, w), the harmonic compensator, 266 Hw.D, harmonic compensator, 254 lA, zero-based invariant subspace, 55 If, invariant subspace generated by /,56 K(z, w), the Bergman kernel, Ka(z, w), the Bergman kernel for standard weights, Kw(z, w), weighted Bergman kernel, 245 N(r), counting function, 100 P(z, w), Poisson kernel, 29 P(z, w), the Poisson kernel, 114,217 P*, the sweep operator, 87 Qw(z, w), weighted harmonic Bergman kernel, 252 13, the Bloch space, 13 130, the little Bloch space, 13 C, the complex plane, lDJ, the open unit disk, 6., the Laplacian, 32 fez, w), biharmonic Green function, 62 Index r[n, biharmonic Green potential, 62 r (J), the weighted biharmonic Green function, 242 r a(Z), approximate Stolz angle, 111 JR, the real line, T, the unit circle, A-oo, the big growth space, 110 A -a, standard growth space, 110 A+a, limit of growth spaces, 112 A=a, limit of growth spaces, 112 Aoa, small growth space, 110, 137 A P, the Bergman spaces, A P -outer function, 190, 191 A~, the standard weighted Bergman space, A~-inner function, 52 A~-subinner function, 87 A~, the Bergman-Nevanlinna class, 98, 131 A2(w), weighted Bergman space, 245 A2(w)-inner function, 270 B, the Berezin transform, 29 Ba, the weighted Berezin transform, 29 {3, the hyperbolic metric, 16 V, the Dirichlet space, 25 C(]),13 Co(IDl), 13 oCT), diagonal, 253 HP2(w),251 MO(f), "mean oscillation", 43 MO r (f), mean oscillation, 42 H(IDl), the space of all holomorphic functions, 18 Hoo,2 KB+,194 K·-absolutely continuous, 201 K·-area, 148 K·-bound,194 K·-bounded above, 194 K·-density, 105 K -smooth, 201 K-variation, 194 A, invariant Laplacian, 32 V, the gradient, 51 283 w, harmonic measure, 218 w, weight, 242, 244 w-mean value disk, 255 ~(A, E), partial Blaschke sum on E, 105 A(A, E), partial logarithmic Blaschke sum on E, 105 P, Bergman projection, Pa, weighted Bergman projection, p, the pseudohyperbolic metric, 16 ~,2 Stolz star, 105 Stolz angle with aperture ct, 106 Sz, Stolz angle, 104 rpz, the Mobius involution, W"r, harmonic measure, 255 KCF), Beurling-Carleson characteristic, 104 dA, the normalized area element, dAa, the weighted area element, dAE, push-out measure, 114 dc, Euclidean metric on the plane, 105 d1f, circle metric, 104 ds, normalized arc length measure, 59, 104 nCr), counting function, 100 n I , common multiplicity of zero at the origin, 56 BMOa,45 BMO r ,42 VMOa,46 VMO r ,46 "inner-outer" factorization, 78, 186, 239 SF, Sz,a, absolutely continuous measure, 201 admissible, 122 analytic projection, 12 aperture, 104, 106 arc length, 104 arc length normalized, 104 arithmetic-geometric mean inequality, 101 284 Index asymptotic K-density lower, 105 upper, 105 atomic singularity, 59 balayage-type estimate, 112 Berezin transform, 28 Bergman kernel, Bergman metric, 16 Bergman projection, Bergman space, Bergman-Nevanlinna class, 98, 131 Besov space, 24, 50 Beurling's theorem, 176 Beurling-Carleson characteristic, 104 Beurling-Carleson set, 104,200 Beurling-type theorem, 181 bi-harmonic Green function, 62 bi-harmonic Green potential, 62 biharmonic Green function, 59, 242 Blaschke product, 26 Bloch space, 13 BMO,42 BMO in the Bergman metric, 42 BMOA,13 bounded K-variation, 194 building blocks, 219 Caratheodory-Schur theorem, 87 Carleson measure, 38 Carleson square, 106, 240 Cayley transform, 113, 165, 167 complementary arcs, 104 concave operator, 182 concave sequence, 183 concavity-type property of the biharmonic Green function, 273 conjugate exponent, 18, 160 contractive divisibility property, 59 contractive mUltiplier, 54 contractive zero divisors, 71 covering, 125,204 cyclic function, 190 cyclic invariant subspace, 56 cyclic vector, 78,85, 190 density, 104 Dirichlet space, 25 division property, 189 domination, 190, 191 duality theorem of Linear Programming, 124,204 edge point in polyhedron, 124 eigenfunctions, 36 eigenvalue, 37 elementary factors, 131 entropy, 104 exact type, 111 expansive multiplier property, 59, 66, 70 extraneous zero, 72 extremal function, 57 extremal problem, 26, 55, 56, 195 extremal problem for I, 56 factorization, 78 Fejer kernel, 87 fractional differentiation, 18 fractional integration, 18 functions of bounded variation, 195 gap sequence, 15 Garsia's lemma, 42 generation, 233 geometric progression, 107 Green function, 59 Green potential, 60 Green's formula, 59 growth space, 110 Hadamard's variational formula, 255, 257 Hankel operator, 49 Hardy space, 12,54,78 harmonic compensator, 254, 266 harmonic conjugate, 26 harmonic majorant, 55 Index harmonic measure, 218, 219, 225, 232 Helly selection theorem, 195 Herglotz measure, 201 Herg10tz transform, 259 homogeneous decomposition, 178 Hurwitz's theorem, 77 hyperbolic center, 42 hyperbolic disk, 38 hyperbolic exponential type, 111 hyperbolic metric, 16 hyperbolic radius, 42 index, 176 index n, 176 index of an invariant subspace, 176 inner function, 52, 78 inner function for Bergman spaces, 53 inner space, 180 inner-outer factorization, 78 interpolating sequence, 50, 136 interpolation problem, 137 invariant Laplacian, 32 invariant subspace, 55 invariant subspace problem, 187 iteration scheme, 222 Jensen's formula, 98 Jensen-type inequality, 116 Krebe function, 260 lacunary series, 15 Laplace equation, 62 Laplace-Beltrami operator, 33 Laplacian, 32 Laplacian invariant, 32 Linear Programming, 119 Linear Programming duality theorem, 124, 204 little Bloch space, 13 localization trick, 230 logarithmic entropy, 194 logarithmically sUbharmonic, 94, 242, 244 285 lower asymptotic K-density, 105 lower Seip density, 153 lunula,227 Mobius group, Mobius map, Mobius transformation, 16 maximal inner space, 187 mean oscillation, 42 min-max equation, 123 minimal type, 111 mushroom, 231 mushroom forest, 230 mushroom hat, 232 mushroom stem, 231 nonoverlapping arcs, 211 normalized arc length, 59, 104 oblique projection, 119 optimization problem, 123 outer function, 190 perturbation, 139 Poincare metric, 16 point-evaluation, Poisson extension, 114 Poisson formula, 29 Poisson kernel, 29, 114 Poisson solver, 273 Poisson transform, 28, 29, 114,273 positive function, measure, premeasure, 190, 193 pseudohyperbolic metric, 16 push-out measure, 114 quasi-Banach space, quasi-similar operators, 188 regular sequence, 177 reproducing for the origin, 242 reproducing kernel, 57, 242 residue theorem, 76 286 Index restriction operator, 155 reverse triangle inequality, 177 sampling sequence, 136 Schur's test, 9, 11 Seip density lower, 153 Seip density upper, 147 separated sequence, 39,50, 138 sequence of interpolation, 136 sequence of sampling, 136 sesquiholomorphic, 252 similar operators, 188 simple covering, 125,204 singly generated invariant subspace, 56 singular inner function, 59 singular measure, 201 spectral mapping theorem, 37 Stieltjes integral, 195 Stolz angle, 104 Stolz star, 105 strictly positive, subinner function, 86 suppressive weight, 273 sweep of a function, 88, 264 Toeplitz operator, 49 truncation, 202 type, hyperbolic exponential, III uniform separating upper asymptotic K-density, 140 uniform upper asymptotic K-*-density, 146 uniformly discrete, 39 upper asymptotic K-density, 105 upper Seip density, 147 variational argument, 56 variational formula, 255 VMO,42 weakly cyclic, 201 Weierstrass factorization, 131 weighted Bergman kernel, weighted Bergman projection, weighted Bergman space, weighted biharmonic Green function, 242 weighted Hele-Shaw flows, 255 Wirtinger derivatives, zero divisor, 57 zero sequence, 98 zero set, 98 zero-based invariant subspace, 56 (continued from page ii) 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 18 79 80 811 82 83 84 85 86 87 88 89 90 91 92 KARGA1'oLOv/MERLZJAKOV Fundamentals of the Theory of Groups BOLLOBAS Graph Theory EDWARDS Fourier Series Vol I 2nd ed WELLS Differential Analysis on Complex Manifolds 2nd ed WATERHOUSE Introduction to Affine Group Schemes SERRE Local Fields WEIDMANN Linear Operators in Hilbert Spaces LANG Cyclotomic Fields II MASSEY Singular Homology Theory F ARKAslKRA Riemann Surfaces 2nd ed STILLWELL Classical Topology and Combinatorial 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WHITEHEAD Elements of Homotopy (continued after index) Haakan Hedenmalm Boris Korenblum Kehe Zhu Theory of Bergman Spaces With Illustrations Springer Haakan Hedenmalm Department of Mathematics Lund... (2000): 47-01, 47A15, 32A30 Library of Congress Cataloging-in-Publication Data Hedenmalm, Haakan Theory of Bergman spaces I Haakan Hedenmalrn, Boris Korenblurn, Kehe Zhu p cm - (Graduate texts in rnathernatics... the subspace of B consisting of functions I with lim (1 Izl-+l- -ld)I/'(z)1 = o The Bloch space plays the same role in the theory of Bergman space as the space BMOA does in the theory of Hardy spaces