Exposedness in Hardy spaces of domains of finite connectivity. (English) Zbl 1012.30023
The Hardy spaces \(H^1(\Omega)\) over domains of finite connectivity are under consideration in the present paper. The boundary properties of the unit ball of \(H^1(\Omega)\), well understood in the classical case \(\Omega=D\), the unit disk, are studied for general multiply connected domains. The authors show that most of the “easier” results for extreme and exposed points in \(H^1(D)\) can be carried over to \(H^1(\Omega)\), whereas some of the “deeper”ones do not hold for the general case. For instance, the extreme points in \(H^1(D)\) are exactly outer functions of norm 1, but there are extreme functions in \(H^1(\Omega)\), that are not outer.
The authors prove that the extreme functions \(f\in H^1(\Omega)\) such that \(|f|^{-1}\) is integrable with respect to the arc length on the boundary are exposed functions. The examples of extreme but not exposed points in \(H^1(\Omega)\) are given for special domains (real slit domains) \(\Omega\).
The authors prove that the extreme functions \(f\in H^1(\Omega)\) such that \(|f|^{-1}\) is integrable with respect to the arc length on the boundary are exposed functions. The examples of extreme but not exposed points in \(H^1(\Omega)\) are given for special domains (real slit domains) \(\Omega\).
Reviewer: Leonid Golinskii (Kharkov)
MSC:
| 30D55 | \(H^p\)-classes (MSC2000) |
| 46E15 | Banach spaces of continuous, differentiable or analytic functions |
| 46B20 | Geometry and structure of normed linear spaces |
Keywords:
domains of finite connectivity; extreme points; exposed points; rigid functions; inner-outer factorizationReferences:
| [1] | Beneker, P., Strongly exposed points, Helson-Szegö weights and Toeplitz operators, J. Int. Eq. Oper. Th., 31, 299-306 (1998) · Zbl 0910.30031 |
| [2] | Duren, P. L., Theory of \(H^p\)-spaces (1970), Academic Press · Zbl 0215.20203 |
| [3] | Fisher, S. D., Function Theory On Planar Domains (1983), John Wiley & Sons · Zbl 0511.30022 |
| [4] | Forelli, F., Extreme points in \(H^1(R)\), Can. J. Math., 19, 312-320 (1967) · Zbl 0155.18502 |
| [5] | Gamelin, T. W.; Voichick, M., Extreme points in spaces of analytic functions, Can. J. Math., 20, 919-928 (1968) · Zbl 0159.41902 |
| [6] | Garnett, J. B., Bounded analytic functions (1981), Academic Press: Academic Press San Diego · Zbl 0469.30024 |
| [7] | Inoue, J., An example of a non-exposed extreme function in the unit ball of \(H^1\), Proc. Edinburgh Math. Soc., 37, 47-51 (1993) · Zbl 0803.30026 |
| [8] | Koosis, P., Introduction to \(H^p\) spaces (1980), Cambridge University Press · Zbl 0435.30001 |
| [9] | Leeuw, K.de; W, Rudin, Extreme points and extremum problems in \(H^1\), Pacific J. Math., 8, 467-485 (1958) · Zbl 0084.27503 |
| [10] | Nakazi, T., Exposed points and extremal problems in \(H^1\), J. Funct. Anal., 53, 224-230 (1983) · Zbl 0527.46049 |
| [11] | Nakazi, T., Exposed points and extremal problems in \(H^1\), II. Tôhoku Math. J., 37, 265-269 (1985) · Zbl 0584.46041 |
| [12] | Nakazi, T. - Personal communication.; Nakazi, T. - Personal communication. |
| [13] | Nehari, Z., Conformal Mapping (1952), McGraw-Hill · Zbl 0048.31503 |
| [14] | Newman, D. J., Pseudo-uniform convexity in \(H^1\), Proc. Amer. Math. Soc., 14, 676-679 (1963) · Zbl 0118.10602 |
| [15] | Parreau, M., Sur les moyennes des fonctions harmoniques et analytiques et la classification des surfaces Riemann, Ann. Inst. Fourier (Grenoble), 3, 103-197 (1951) · Zbl 0047.32004 |
| [16] | Sarason, D. E., Exposed points in \(H^1\), (I. Operator Theory: Advances and Applications, 41 (1989), Birkhäuser Verlag: Birkhäuser Verlag Basel), 485-496 · Zbl 0678.46019 |
| [17] | Sarason, D. E., Exposed points in \(H^1\), (II. Operator Theory: Advances and Applications, 48 (1990), Birkhäuser Verlag: Birkhäuser Verlag Basel), 333-347 · Zbl 0744.46042 |
| [18] | Temme, D.; J, Wiegerinck, Extremal properties of the unit ball of \(H^1\), Indag. Mathem., N.S., 3, 1, 119-127 (1992) · Zbl 0757.46031 |
| [19] | Voichick, M.; L, Zalcman, Inner and outer functions on Riemann surfaces, Proc. Amer. Math. Soc., 16, 1200-1204 (1965) · Zbl 0147.11503 |
| [20] | Wiegerinck, J., A characterization of strongly exposed points of the unit ball of \(H^1\), Indag. Mathem., N.S., 4, 4, 509-519 (1993) · Zbl 0802.32015 |
| [21] | Yabuta, K., Unicity of the extremum problems in \(H^1\) (\(U^n\)), Proc. Amer. Math. Soc., 28, 181-184 (1971) · Zbl 0219.32002 |
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