Analysis & PDE

  • Anal. PDE
  • Volume 4, Number 4 (2011), 499-550.

The corona theorem for the Drury–Arveson Hardy space and other holomorphic Besov–Sobolev spaces on the unit ball in $\C^n$

Şerban Costea, Eric Sawyer, and Brett Wick

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Abstract

We prove that the multiplier algebra of the Drury–Arveson Hardy space Hn2 on the unit ball in n has no corona in its maximal ideal space, thus generalizing the corona theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov–Sobolev space Bpσ has the “baby corona property” for all σ0 and 1<p<. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems.

Article information

Source
Anal. PDE, Volume 4, Number 4 (2011), 499-550.

Dates
Received: 10 March 2010
Revised: 25 May 2010
Accepted: 23 June 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731167

Digital Object Identifier
doi:10.2140/apde.2011.4.499

Mathematical Reviews number (MathSciNet)
MR3077143

Zentralblatt MATH identifier
1264.32004

Subjects
Primary: 30H05: Bounded analytic functions 32A37: Other spaces of holomorphic functions (e.g. bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) [See also 46Exx]

Keywords
Besov–Sobolev Spaces corona Theorem several complex variables Toeplitz corona theorem

Citation

Costea, Şerban; Sawyer, Eric; Wick, Brett. The corona theorem for the Drury–Arveson Hardy space and other holomorphic Besov–Sobolev spaces on the unit ball in $\C^n$. Anal. PDE 4 (2011), no. 4, 499--550. doi:10.2140/apde.2011.4.499. https://projecteuclid.org/euclid.apde/1513731167


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References

  • J. Agler and J. E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics 44, American Mathematical Society, Providence, RI, 2002.
  • É. Amar, “On the corona problem”, J. Geom. Anal. 1:4 (1991), 291–305.
  • E. Amar, “On the Toëplitz corona problem”, Publ. Mat. 47:2 (2003), 489–496.
  • C.-G. Ambrozie and D. Timotin, “On an intertwining lifting theorem for certain reproducing kernel Hilbert spaces”, Integral Equations Operator Theory 42:4 (2002), 373–384.
  • M. Andersson, “The $H\sp 2$ corona problem and $\overline\partial\sb b$ in weakly pseudoconvex domains”, Trans. Amer. Math. Soc. 342:1 (1994), 241–255.
  • M. E. L. Andersson, “On the $H\sp p$ corona problem”, Bull. Sci. Math. 118:3 (1994), 287–306.
  • M. Andersson and H. Carlsson, “Wolff type estimates and the $H\sp p$ corona problem in strictly pseudoconvex domains”, Ark. Mat. 32:2 (1994), 255–276.
  • M. Andersson and H. Carlsson, “Estimates of solutions of the $H\sp p$ and BMOA corona problem”, Math. Ann. 316:1 (2000), 83–102.
  • M. Andersson and H. Carlsson, “$Q\sb p$ spaces in strictly pseudoconvex domains”, J. Anal. Math. 84 (2001), 335–359.
  • N. Arcozzi, R. Rochberg, and E. Sawyer, Carleson measures and interpolating sequences for Besov spaces on complex balls, Mem. Amer. Math. Soc. 859, American Mathematical Society, Providence, RI, 2006.
  • N. Arcozzi, R. Rochberg, and E. Sawyer, “Carleson measures for the Drury–Arveson Hardy space and other Besov–Sobolev spaces on complex balls”, Adv. Math. 218:4 (2008), 1107–1180.
  • W. Arveson, “Subalgebras of $C\sp *$-algebras, III: Multivariable operator theory”, Acta Math. 181:2 (1998), 159–228.
  • J. A. Ball, T. T. Trent, and V. Vinnikov, “Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces”, pp. 89–138 in Operator theory and analysis (Amsterdam, 1997), edited by H. Bart et al., Oper. Theory Adv. Appl. 122, Birkhäuser, Basel, 2001.
  • F. Beatrous, Jr., “Estimates for derivatives of holomorphic functions in pseudoconvex domains”, Math. Z. 191:1 (1986), 91–116.
  • L. Carleson, “Interpolations by bounded analytic functions and the corona problem”, Ann. of Math. $(2)$ 76 (1962), 547–559.
  • P. Charpentier, “Formules explicites pour les solutions minimales de l'équation $\bar \partial u=f$ dans la boule et dans le polydisque de ${\bf C}\sp{n}$”, Ann. Inst. Fourier $($Grenoble$)$ 30:4 (1980), 121–154.
  • Ş. Costea, E. T. Sawyer, and B. D. Wick, “BMO estimates for the $H\sp \infty(\mathbb B\sb n)$ Corona problem”, J. Funct. Anal. 258:11 (2010), 3818–3840.
  • P. A. Fuhrmann, “On the corona theorem and its application to spectral problems in Hilbert space”, Trans. Amer. Math. Soc. 132 (1968), 55–66.
  • J. B. Garnett, Bounded analytic functions, Pure and Applied Mathematics 96, Academic Press, New York, 1981.
  • S. G. Krantz and S.-Y. Li, “Some remarks on the corona problem on strongly pseudoconvex domains in ${\bf C}\sp n$”, Illinois J. Math. 39:2 (1995), 323–349.
  • E. Ligocka, “Estimates in Sobolev norms $\Vert \cdot\Vert \sp s\sb p$ for harmonic and holomorphic functions and interpolation between Sobolev and Hölder spaces of harmonic functions”, Studia Math. 86:3 (1987), 255–271.
  • K.-C. Lin, “The $H\sp p$-corona theorem for the polydisc”, Trans. Amer. Math. Soc. 341:1 (1994), 371–375. http:www.ams.org/mathscinet-getitem?mr=94c:46106MR 94c:46106
  • N. K. Nikolski, Operators, functions, and systems: an easy reading, vol. 1: Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs 92, American Mathematical Society, Providence, RI, 2002.
  • J. M. Ortega and J. Fàbrega, “Pointwise multipliers and decomposition theorems in analytic Besov spaces”, Math. Z. 235:1 (2000), 53–81.
  • J. Ortega and J. Fàbrega, “Multipliers in Hardy–Sobolev spaces”, Integral Equations Operator Theory 55:4 (2006), 535–560.
  • N. Øvrelid, “Integral representation formulas and $L\sp{p}$-estimates for the $\bar \partial $-equation”, Math. Scand. 29 (1971), 137–160.
  • M. Rosenblum, “A corona theorem for countably many functions”, Integral Equations Operator Theory 3:1 (1980), 125–137.
  • W. Rudin, Function theory in the unit ball of ${\bf C}\sp{n}$, Grundlehren der Mathematischen Wissenschaften 241, Springer, New York, 1980.
  • E. T. Sawyer, Function theory: interpolation and corona problems, Fields Institute Monographs 25, American Mathematical Society, Providence, RI, 2009.
  • E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, Princeton University Press, Princeton, NJ, 1993.
  • V. A. Tolokonnikov, “Estimates in Carleson's corona theorem and finitely generated ideals in an $H\sp{\infty }$ algebra”, Funktsional. Anal. i Prilozhen. 14:4 (1980), 85–86. In Russian; translated in Funct. Anal. Appl. \bf14:4 (1981), 320–322.
  • V. A. Tolokonnikov, “Estimates in the Carleson corona theorem, ideals of the algebra $H\sp{\infty }$, a problem of Szökefalvi-Nagy”, 113 (1981), 178–198. In Russian; translated in J. Sov. Math. 22:6 (1983), 1814–1828. ISSN 0090-4104.
  • V. A. Tolokonnikov, “The corona theorem in algebras of bounded analytic functions”, pp. 61–93 in Thirteen papers in algebra, functional analysis, topology, and probability, Amer. Math. Soc. Trans. 149, 1991.
  • S. R. Treil', “Angles between co-invariant subspaces, and the operator corona problem: a question of Szőkefalvi-Nagy”, Dokl. Akad. Nauk SSSR 302:5 (1988), 1063–1068. In Russian; translated in Soviet Math. Dokl. 38:2 (1989), 394-399.
  • S. Treil and B. D. Wick, “The matrix-valued $H\sp p$ corona problem in the disk and polydisk”, J. Funct. Anal. 226:1 (2005), 138–172.
  • T. T. Trent, “A corona theorem for multipliers on Dirichlet space”, Integral Equations Operator Theory 49:1 (2004), 123–139.
  • T. T. Trent, “An $H\sp 2$-corona theorem on the bidisk for infinitely many functions”, Linear Algebra Appl. 379 (2004), 213–227.
  • T. T. Trent and B. D. Wick, “Toeplitz corona theorems for the polydisk and the unit ball”, Complex Anal. Oper. Theory 3:3 (2009), 729–738.
  • T. Trent and X. Zhang, “A matricial corona theorem”, Proc. Amer. Math. Soc. 134:9 (2006), 2549–2558.
  • N. T. Varopoulos, “BMO functions and the $\overline \partial $-equation”, Pacific J. Math. 71:1 (1977), 221–273.
  • J. Xiao, “The $\overline\partial$-problem for multipliers of the Sobolev space”, Manuscripta Math. 97:2 (1998), 217–232. http:www.ams.org/mathscinet-getitem?mr=99g:46047MR 99g:46047
  • K. Zhu, Spaces of holomorphic functions in the unit ball, Graduate Texts in Mathematics 226, Springer, New York, 2005.

Supplemental materials

  • PDF file containing proofs of formulas and modifications of arguments already in the literature that would otherwise interrupt the main flow of the paper.