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  1. Home
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  3. A Primer on the Dirichlet Space
  • Cited by 111
Publisher:
Cambridge University Press
Online publication date:
January 2014
Print publication year:
2014
Online ISBN:
9781107239425
DOI:
https://doi.org/10.1017/CBO9781107239425
Subjects:
Mathematics (general), Mathematics, Abstract Analysis, Real and Complex Analysis
Series:
Cambridge Tracts in Mathematics (203)
Information

Book description

The Dirichlet space is one of the three fundamental Hilbert spaces of holomorphic functions on the unit disk. It boasts a rich and beautiful theory, yet at the same time remains a source of challenging open problems and a subject of active mathematical research. This book is the first systematic account of the Dirichlet space, assembling results previously only found in scattered research articles, and improving upon many of the proofs. Topics treated include: the Douglas and Carleson formulas for the Dirichlet integral, reproducing kernels, boundary behaviour and capacity, zero sets and uniqueness sets, multipliers, interpolation, Carleson measures, composition operators, local Dirichlet spaces, shift-invariant subspaces, and cyclicity. Special features include a self-contained treatment of capacity, including the strong-type inequality. The book will be valuable to researchers in function theory, and with over 100 exercises it is also suitable for self-study by graduate students.

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Contents

Contents
References
References
[1] Adams, D. R., and Hedberg, L. I. 1996. Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften, vol. 314. Berlin Google Scholar: Springer-Verlag.
[2] Agler, J. 1988. Some interpolation theorems of Nevanlinna-Pick type Google Scholar. Unpublished manuscript.
[3] Agler, J. 1990. A disconjugacy theorem for Toeplitz operators. Amer. J. Math., 112 Google Scholar(1), 1–14.
[4] Agler, J., and McCarthy, J. E. 2002. Pick Interpolation and Hilbert Function Spaces. Graduate Studies in Mathematics, vol. 44. Providence, RI Google Scholar: American Mathematical Society.
[5] Ahlfors, L. V. 1973. Conformai Invariants: Topics in Geometric Function Theory. McGraw-Hill Series in Higher Mathematics. New York Google Scholar: McGraw-Hill.
[6] Aikawa, H., and Essen, M. 1996. Potential Theory—Selected Topics. Lecture Notes in Mathematics, vol. 1633. Berlin Google Scholar: Springer-Verlag.
[7] Aleman, A. 1993. The multiplication operator on Hilbert spaces of analytic functions. Habilitationsschrift, Fern Universität, Hagen Google Scholar.
[8] Arazy, J., and Fisher, S. D. 1985. The uniqueness of the Dirichlet space among Mobius-invariant Hilbert spaces. Illinois J. Math., 29 Google Scholar(3), 449–462.
[9] Arcozzi, N., Rochberg, R., and Sawyer, E. T. 2002. Carleson measures for analytic Besov spaces. Rev. Mat. Iberoamericana, 18 Google Scholar(2), 443–510.
[10] Arcozzi, N., Rochberg, R., and Sawyer, E. T. 2008. Carleson measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on complex balls. Adv. Math., 218 Google Scholar(4), 1107–1180.
[11] Arcozzi, N., Rochberg, R., Sawyer, E. T., and Wick, B. D. 2011. The Dirichlet space: a survey. New York J. Math., 17A Google Scholar, 45–86.
[12] Armitage, D. H., and Gardiner, S. J. 2001. Classical Potential Theory. Springer Monographs in Mathematics. London Google Scholar: Springer-Verlag.
[13] Bénéteau, C., Condori, A., Liaw, C., Seco, D., and Sola, A.Cyclicity in Dirichlet-type spaces and extremal polynomials. J. Anal. Math. Google Scholar To appear.
[14] Berenstein, C. A., and Gay, R. 1991. Complex Variables. Graduate Texts in Mathematics, vol. 125. New York Google Scholar: Springer-Verlag.
[15] Beurling, A. 1933. Etudes sur un problème de majoration Google Scholar. Doctoral thesis, Uppsala University.
[16] Beurling, A. 1940. Ensembles exceptionnels. Acta Math., 72 Google Scholar, 1–13.
[17] Bishop, C. J. 1994. Interpolating sequences for the Dirichlet space and its multipliers Google Scholar. Unpublished manuscript.
[18] Bøe, B. 2005. An interpolation theorem for Hilbert spaces with Nevanlinna-Pick kernel. Proc. Amer. Math. Soc., 133 Google Scholar(7), 2077–2081.
[19] Bogdan, K. 1996. On the zeros of functions with finite Dirichlet integral. Kodai Math. J., 19 Google Scholar(1), 7–16.
[20] Borichev, A. 1993. A note on Dirichlet-type spaces Google Scholar. Uppsala University Department of Mathematics Report 11.
[21] Borichev, A. 1994. Boundary behavior in Dirichlet-type spaces Google Scholar. Uppsala University Department of Mathematics Report 3.
[22] Bourdon, P. S. 1986. Cellular-indecomposable operators and Beurling's theorem. Michigan Math J., 33 Google Scholar(2), 187–193.
[23] Brown, L., and Cohn, W. 1985. Some examples of cyclic vectors in the Dirichlet space. Proc. Amer. Math. Soc., 95 Google Scholar(1), 42–46.
[24] Brown, L., and Shields, A. L. 1984. Cyclic vectors in the Dirichlet space. Trans. Amer. Math. Soc., 285 Google Scholar(1), 269–303.
[25] Carleson, L. 1952a. On the zeros of functions with bounded Dirichlet integrals. Math. Z., 56 Google Scholar, 289–295.
[26] Carleson, L. 1952b. Sets of uniqueness for functions regular in the unit circle. Acta Math., 87 Google Scholar, 325–345.
[27] Carleson, L. 1960. A representation formula for the Dirichlet integral. Math. Z., 73 Google Scholar, 190–196.
[28] Carleson, L. 1962. Interpolations by bounded analytic functions and the corona problem. Ann. Math. (2), 76 Google Scholar, 547–559.
[29] Carleson, L. 1967. Selected Problems on Exceptional Sets. Van Nostrand Mathematical Studies, No. 13. Princeton, NJ Google Scholar: Van Nostrand.
[30] Carlsson, M. 2008. On the Cowen-Douglas class for Banach space operators. Integral Equations Operator Theory, 61 Google Scholar(4), 593–598.
[31] Caughran, J. G. 1969. Two results concerning the zeros of functions with finite Dirichlet integral. Canad. J. Math., 21 Google Scholar, 312–316.
[32] Caughran, J. G. 1970. Zeros of analytic functions with infinitely differentiable boundary values. Proc. Amer. Math. Soc., 24 Google Scholar, 700–704.
[33] Chacón, G. R. 2011. Carleson measures on Dirichlet-type spaces. Proc. Amer. Math. Soc., 139 Google Scholar(5), 1605–1615.
[34] Chang, S.-Y. A., and Marshall, D. E. 1985. On a sharp inequality concerning the Dirichlet integral. Amer. J. Math., 107 Google Scholar(5), 1015–1033.
[35] Chartrand, R. 2002. Toeplitz operators on Dirichlet-type spaces. J. Operator Theory, 48 Google Scholar(1), 3–13.
[36] Cowen, C. C., and MacCluer, B. D. 1995. Composition Operators on Spaces of Analytic Functions. Studies in Advanced Mathematics. Boca Raton, FL Google Scholar: CRC Press.
[37] Cowen, M. J., and Douglas, R. G. 1978. Complex geometry and operator theory. Acta Math., 141 Google Scholar(3–4), 187–261.
[38] Doob, J. L. 1984. Classical Potential Theory and its Probabilistic Counterpart. Grundlehren der Mathematischen Wissenschaften, vol. 262. New York Google Scholar: Springer-Verlag.
[39] Douglas, J. 1931. Solution of the problem of Plateau. Trans. Amer. Math. Soc., 33 Google Scholar(1), 263–321.
[40] Duren, P. L. 1970. Theory of Hp Spaces. Pure and Applied Mathematics, Vol. 38. New York Google Scholar: Academic Press.
[41] Dyn′kin, E. M. 1972. Extensions and integral representations of smooth functions of one complex variable. Dissertation, Leningrad Google Scholar.
[42] El-Fallah, O., Kellay, K., Mashreghi, J., and Ransford, T.One-box conditions for Carleson measures for the Dirichlet space. Proc. Amer. Math. Soc. Google Scholar To appear.
[43] El-Fallah, O., Kellay, K., and Ransford, T. 2006. Cyclicity in the Dirichlet space. Ark. Mat., 44 Google Scholar(1), 61–86.
[44] El-Fallah, O., Kellay, K., and Ransford, T. 2009. On the Brown-Shields conjecture for cyclicity in the Dirichlet space. Adv. Math., 222 Google Scholar(6), 2196–2214.
[45] El-Fallah, O., Kellay, K., Shabankhah, M., and Youssfi, H. 2011. Level sets and composition operators on the Dirichlet space. J. Funct. Anal., 260 Google Scholar(6), 1721–1733.
[46] El-Fallah, O., Kellay, K., Mashreghi, J., and Ransford, T. 2012. A self-contained proof of the strong-type capacitary inequality for the Dirichlet space. Pages 1–20 of: Complex Analysis and Potential Theory. CRM Proc. Lecture Notes, vol. 55. Providence, RI Google Scholar: American Mathematical Society.
[47] Essén, M. 1987. Sharp estimates of uniform harmonic majorants in the plane. Ark. Mat., 25 Google Scholar(1), 15–28.
[48] Frostman, O. 1935. Potentiel d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Thesis. Meddel. Lunds Univ. Mat. Sem., 3 Google Scholar, 1–118.
[49] Gallardo-Gutiérrez, E. A., and González, M. J. 2003. Exceptional sets and Hilbert-Schmidt composition operators. J. Funct. Anal., 199 Google Scholar(2), 287–300.
[50] Garnett, J. B. 2007. Bounded Analytic Functions. revised first edn. Graduate Texts in Mathematics, vol. 236. New York Google Scholar: Springer.
[51] Garnett, J. B., and Marshall, D. E. 2005. Harmonic Measure. New Mathematical Monographs, vol. 2. Cambridge Google Scholar: Cambridge University Press.
[52] Guillot, D. 2012a. Blaschke condition andzero sets in weighted Dirichlet spaces. Ark. Mat., 50 Google Scholar(2), 269–278.
[53] Guillot, D. 2012b. Fine boundary behavior and invariant subspaces of harmonically weighted Dirichlet spaces. Complex Anal. Operator Theory, 6 Google Scholar(6), 1211–1230.
[54] Hansson, K. 1979. Imbedding theorems of Sobolev type in potential theory. Math. Scand., 45 Google Scholar(1), 77–102.
[55] Hardy, G. H. 1949. Divergent Series. Oxford Google Scholar: Clarendon Press.
[56] Hastings, W. W. 1975. A Carleson measure theorem for Bergman spaces. Proc. Amer. Math. Soc., 52 Google Scholar, 237–241.
[57] Hayman, W. K., and Kennedy, P. B. 1976. Subharmonic Functions, vol. 1. London Mathematical Society Monographs, No. 9. London Google Scholar: Academic Press [Harcourt Brace Jovanovich Publishers].
[58] Hedenmalm, H., and Shields, A. L. 1990. Invariant subspaces in Banach spaces of analytic functions. Michigan Math. J., 37 Google Scholar(1), 91–104.
[59] Helms, L. L. 1975. Introduction to Potential Theory. Pure and Applied Mathematics, vol. 22. Huntington, NY Google Scholar: Robert E. Krieger.
[60] Hille, E. 1962. Analytic Function Theory, vol. II. Introductions to Higher Mathematics. Boston, MA Google Scholar: Ginn and Co.
[61] Kahane, J.-P., and Salem, R. 1994. Ensembles parfaits et séries trigono-métriques. Second edn. Paris Google Scholar: Hermann.
[62] Kellay, K. 2011. Poincaré type inequality for Dirichlet spaces and application to the uniqueness set. Math. Scand., 108 Google Scholar(1), 103–114.
[63] Kellay, K., and Mashreghi, J. 2012. On zero sets in the Dirichlet space. J. Geom. Anal., 22 Google Scholar(4), 1055–1070.
[64] Kerman, R., and Sawyer, E. T. 1988. Carleson measures and multipliers of Dirichlet-type spaces. Trans. Amer. Math. Soc., 309 Google Scholar(1), 87–98.
[65] Koosis, P. 1992. The Logarithmic Integral II. Cambridge Studies in Advanced Mathematics, vol. 21. Cambridge Google Scholar: Cambridge University Press.
[66] Koosis, P. 1998. Introduction to Hp Spaces. Second edn. Cambridge Tracts in Mathematics, vol. 115. Cambridge Google Scholar: Cambridge University Press.
[67] Korenblum, B. I. 1972. Invariant subspaces of the shift operator in a weighted Hilbert space. Math. USSR-Sb., 18 Google Scholar, 111–138.
[68] Korenblum, B. I. 2006. Blaschke sets for Bergman spaces. Pages 145–152 of: Bergman Spaces and Related Topics in Complex Analysis. Contemp. Math., vol. 404. Providence, RI Google Scholar: American Mathematical Society.
[69] Korolevič, V. S. 1970. A certain theorem of Beurling and Carleson. Ukrainian Math. J., 22 Google Scholar(6), 710–714.
[70] Landkof, N. S. 1972. Foundations of Modern Potential Theory. Grundlehren der mathematischen Wissenschaften, vol. 180. New York Google Scholar: Springer-Verlag.
[71] Lefèvre, P., Li, D., Queffélec, H., and Rodríguez-Piazza, L. 2013. Compact composition operators on the Dirichlet space and capacity of sets of contact points. J. Funct. Anal., 264 Google Scholar(4), 895–919.
[72] Luecking, D. H. 1987. Trace ideal criteria for Toeplitz operators. J. Funct. Anal., 73 Google Scholar(2), 345–368.
[73] MacCluer, B. D., and Shapiro, J. H. 1986. Angular derivatives and compact composition operators on the Hardy and Bergman spaces. Canad. J. Math., 38 Google Scholar(4), 878–906.
[74] Malliavin, P. 1977. Sur l'analyse harmonique de certaines classes de séries de Taylor. Pages 71–91 of: Symposia Mathematica, Vol. XXII (Convegno sull'Analisi Armonica e Spazi di Funzioni su Gruppi Localmente Compatti, IN-DAM, Rome, 1976). London Google Scholar: Academic Press.
[75] Marshall, D. E. 1989. A new proof of a sharp inequality concerning the Dirichlet integral. Ark. Mat., 27 Google Scholar(1), 131–137.
[76] Marshall, D. E., and Sundberg, C. 1989. Interpolating sequences for the multipliers ofthe Dirichlet space Google Scholar. Unpublished manuscript.
[77] Mashreghi, J. 2009. Representation Theorems in Hardy Spaces. London Mathematical Society Student Texts, vol. 74. Cambridge Google Scholar: Cambridge University Press.
[78] Mashreghi, J., Ransford, T., and Shabankhah, M. 2010. Arguments of zero sets in the Dirichlet space. Pages 143–148 of: Hilbert Spaces ofAnalytic Functions. CRM Proc. Lecture Notes, vol. 51. Providence, RI Google Scholar: American Mathematical Society.
[79] Maz′ya, V. G., and Havin, V. P. 1973. Application of the (p, l)-capacity to certain problems of the theory of exceptional sets. Mat. Sb. (N.S.), 90 Google Scholar(132), 558–591, 640.
[80] Meyers, N. G. 1970. A theory of capacities for potentials of functions in Lebesgue classes. Math. Scand., 26, 255–292 (1971 Google Scholar).
[81] Monterie, M. A. 1997. Capacities of certain Cantor sets. Indag. Math. (N.S.), 8 Google Scholar(2), 247–266.
[82] Nagel, A., and Stein, E. M. 1984. On certain maximal functions and approach regions. Adv. Math., 54 Google Scholar(1), 83–106.
[83] Nagel, A., Rudin, W., and Shapiro, J. H. 1982. Tangential boundary behavior of functions in Dirichlet-type spaces. Ann. Math. (2), 116 Google Scholar(2), 331–360.
[84] Nagel, A., Rudin, W., and Shapiro, J. H. 1983. Tangential boundary behavior of harmonic extensions of Lp potentials. Pages 533–548 of: Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. I, II (Chicago, IL, 1981). Wadsworth Math. Ser. Belmont, CA Google Scholar: Wadsworth.
[85] Nehari, Z. 1975. Conformai Mapping. New York Google Scholar: Dover Publications.
[86] Novinger, W. P. 1971. Holomorphic functions with infinitely differentiable boundary values. Illinois J. Math., 15 Google Scholar, 80–90.
[87] Ohtsuka, M. 1957. Capacité d'ensembles de Cantor généralisés. Nagoya Math. J., 11 Google Scholar, 151–160.
[88] Olin, R. F., and Thomson, J. E. 1984. Cellular-indecomposable subnormal operators. Integral Equations Operator Theory, 7 Google Scholar(3), 392–430.
[89] Parrott, S. 1978. On a quotient norm and the Sz.-Nagy–Foia lifting theorem. J. Funct. Anal., 30 Google Scholar(3), 311–328.
[90] Peller, V. V., and Hruscev, S. V. 1982. Hankel operators, best approximations and stationary Gaussian processes. Russian Math. Surveys, 37 Google Scholar(1), 61–144.
[91] Pommerenke, Ch. 1975. Univalent Functions. Göttingen Google Scholar: Vandenhoeck & Ruprecht.
[92] Pommerenke, Ch. 1992. Boundary Behaviour ofConformal Maps. Grundlehren der Mathematischen Wissenschaften, vol. 299. Berlin Google Scholar: Springer-Verlag.
[93] Ransford, T. 1995. Potential Theory in the Complex Plane. London Mathematical Society Student Texts, vol. 28. Cambridge Google Scholar: Cambridge University Press.
[94] Ransford, T., and Selezneff, A. 2012. Capacity and covering numbers. Potential Anal., 36 Google Scholar, 223–233.
[95] Richter, S. 1988. Invariant subspaces of the Dirichlet shift. J. Reine Angew. Math., 386 Google Scholar, 205–220.
[96] Richter, S. 1991. A representation theorem for cyclic analytic two-isometries. Trans. Amer. Math. Soc., 328 Google Scholar(1), 325–349.
[97] Richter, S., and Shields, A. L. 1988. Bounded analytic functions in the Dirichlet space. Math. Z., 198 Google Scholar(2), 151–159.
[98] Richter, S., and Sundberg, C. 1991. A formula for the local Dirichlet integral. Michigan Math. J., 38 Google Scholar(3), 355–379.
[99] Richter, S., and Sundberg, C. 1992. Multipliers and invariant subspaces in the Dirichlet space. J. Operator Theory, 28 Google Scholar(1), 167–186.
[100] Richter, S., and Sundberg, C. 1994. Invariant subspaces of the Dirichlet shift and pseudocontinuations. Trans. Amer. Math. Soc., 341 Google Scholar(2), 863–879.
[101] Richter, S., Ross, W. T., and Sundberg, C. 2004. Zeros of functions with finite Dirichlet integral. Proc. Amer. Math. Soc., 132 Google Scholar(8), 2361–2365.
[102] Rochberg, R., and Wu, Z. J. 1992. Toeplitz operators on Dirichlet spaces. Integral Equations Operator Theory, 15 Google Scholar(2), 325–342.
[103] Ross, W. T. 2006. The classical Dirichlet space. Pages 171–197 of: Recent Advances in Operator-Related Function Theory. Contemp. Math., vol. 393. Providence, RI Google Scholar: American Mathematical Society.
[104] Rudin, W. 1987. Real and Complex Analysis. Third edn. New York Google Scholar: McGraw-Hill.
[105] Rudin, W. 1992. Power series with zero-sum on countable sets. Complex Variables Theory Appl., 18 Google Scholar(3–4), 283–284.
[106] Saff, E. B., and Totik, V. 1997. Logarithmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften, vol. 316. Berlin Google Scholar: Springer-Verlag.
[107] Sarason, D. 1986. Doubly shift-invariant spaces in H2. J. Operator Theory, 16 Google Scholar(1), 75–97.
[108] Sarason, D. 1997. Local Dirichlet spaces as de Branges-Rovnyak spaces. Proc. Amer. Math. Soc., 125 Google Scholar(7), 2133–2139.
[109] Seip, K. 2004. Interpolation and Sampling in Spaces of Analytic Functions. University Lecture Series, vol. 33. Providence, RI Google Scholar: American Mathematical Society.
[110] Shapiro, H. S., and Shields, A. L. 1962. On the zeros of functions with finite Dirichlet integral and some related function spaces. Math. Z., 80 Google Scholar, 217–229.
[111] Shapiro, J. H. 1980. Cauchy transforms and Beurling-Carleson-Hayman thin sets. Michigan Math. J., 27 Google Scholar(3), 339–351.
[112] Shapiro, J. H. 1987. The essential norm of a composition operator. Ann. Math. (2), 125 Google Scholar(2), 375–404.
[113] Shapiro, J. H. 1993. Composition Operators and Classical Function Theory. Universitext Google Scholar: Tracts in Mathematics. New York: Springer-Verlag.
[114] Shields, A. L. 1983. An analogue of the Fejér-Riesz theorem for the Dirichlet space. Pages 810–820 of: Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. I, II (Chicago, IL, 1981). Wadsworth Math. Ser. Belmont, CA Google Scholar: Wadsworth.
[115] Shimorin, S. M. 1998. Reproducing kernels and extremal functions in Dirichlet-type spaces. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 255 Google Scholar (Issled. po Linein. Oper. i Teor. Funkts. 26), 198–220, 254. Translation in J. Math. Sci. (New York) 107(4) (2001), 4108–4124.
[116] Shimorin, S. M. 2002. Complete Nevanlinna-Pick property of Dirichlet-type spaces. J. Funct. Anal., 191 Google Scholar(2), 276–296.
[117] Stegenga, D. A. 1980. Multipliers of the Dirichlet space. Illinois J. Math., 24 Google Scholar(1), 113–139.
[118] Taylor, B. A., and Williams, D. L. 1970. Ideals in rings of analytic functions with smooth boundary values. Canad. J. Math., 22 Google Scholar, 1266–1283.
[119] Taylor, G. D. 1966. Multipliers on Dα. Trans. Amer. Math. Soc., 123 Google Scholar, 229–240.
[120] Tsuji, M. 1975. Potential Theory in Modern Function Theory. New York Google Scholar: Chelsea.
[121] Twomey, J. B. 1989. Tangential limits for certain classes of analytic functions. Mathematika, 36 Google Scholar(1), 39–49.
[122] Twomey, J. B. 2002. Tangential boundary behaviour of harmonic and holomorphic functions. J. London Math. Soc. (2), 65 Google Scholar(1), 68–84.
[123] Wojtaszczyk, P. 1991. Banach Spaces for Analysts. Cambridge Studies in Advanced Mathematics, vol. 25. Cambridge Google Scholar: Cambridge University Press.
[124] Wynn, A. 2011. Sufficient conditions for weighted admissibility of operators with applications to Carleson measures and multipliers. Q. J. Math., 62 Google Scholar(3), 747–770.

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