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The pseudoanalytic extension. (English) Zbl 0795.30034
The paper under review is a survey. The main technique is a description of smooth functions in terms of the so-called analytic continuation. The idea of pseudoanalytic continuation was developed by the author. Roughly speaking the approach of pseudoanalytic continuation describes classes of various smooth functions as functions which admit a continuation to a neighbourhood of their domain such that the \(\overline\partial\) derivative is small near the boundary. This approach is shown to be very powerful. The author describes many classes of functions in terms of pseudoanalytic continuation: Hölder classes, Sobolev classes, Besov classes, Carleman classes.
In the survey under review the author demonstrates the technique of pseudoanalytic continuation by solving several problems or giving new proofs: Denjoy-Carleman theorem, interpolation problems in spaces of smooth functions, multipliers properties of inner functions on Hölder classes, and others.
The paper is written very clearly.
Reviewer: V.V.Peller

MSC:
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
26E10 \(C^\infty\)-functions, quasi-analytic functions
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References:
[1] A. A. Borichev, Boundary uniqueness theorems for almost analytic functions and asymmetric algebras of sequences, Math. USSR Sb. 64 (1989), 323–338. · Zbl 0677.30003
[2] A. A. Borichev, Beurling algebras and generalized Fourier transform, LOMI preprint, E-15-88, Leningrad, 1989, 54pp.
[3] A. A. Borichev and A. L. Volberg, Uniqueness theorems for almost analytic functions, Leningrad Math. J. 1 (1990), 157–191. · Zbl 0725.30038
[4] J. E. Brennan and A. L. Volberg, Asymptotic values and the growth of analytic functions in spiral domains, to appear. · Zbl 0798.30021
[5] J. Bruna, Les ensembles d’interpolation des A p (D), C. R. Acad. Sci. Paris 290 (1980), A25–27. · Zbl 0439.30026
[6] L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1959), 921–930. · Zbl 0085.06504
[7] K. M. Dyakonov, Division and multiplication by inner functions and embedding theorems for star-invariant subspaces, Amer. J. Math., to appear. · Zbl 0802.30030
[8] E. M. Dyn’kin, An operator calculus based on the Cauchy-Green formula, Sem. Math. Steklov Math. Iust. Leningrad 19 (1970), 221–226.
[9] E. M. Dyn’kin, Functions with given estimate for $§rtial f/§rtial { z}$ and N. Levinson’s theorem, Math. USSR. Sb. 18 (1972).
[10] E. M. Dyn’kin, On the growth of analytic function near its singularity set, J. Soviet Math. 4 (1975).
[11] E. M. Dyn’kin, Estimates for analytic functions in Jordan domains, Zap. Naucn. Semin. LOMI 73 (1977), 70–90. · Zbl 0407.30029
[12] E. M. Dyn’kin, Free interpolation sets for Hölder classes. Math. USSR Sb. 37 (1980), 97–117. · Zbl 0433.30030
[13] E. M. Dyn’kin, On the smoothness of integrals of Cauchy type, Soviet Math, Dokl. 21 (1980), 199–202. · Zbl 0449.47045
[14] E. M. Dyn’kin, Pseudoanalytic extensions of smooth functions. The uniform scale, Am. Math. Soc. Transl. (2) 115 (1980), 33–58.
[15] E. M. Dyn’kin, The rate of polynomial approximation in the complex domain, Springer Lecture Notes in Math. 864 (1981), 90–142.
[16] E. M. Dyn’kin, The constructive characterization of Sobolev and Besov classes, Proc. Steklov Math. Inst. 155 (1983), 39–74. · Zbl 0512.46037
[17] E. M. Dyn’kin, Free interpolation by functions with H 1 derivative, J. Soviet Math. 27 (1984), 77–87.
[18] E. M. Dyn’kin, Methods of the theory of singular integrals II, Encyclopaedia Math. Sci. Vol. 42, Springer, Berlin, 1991.
[19] J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. · Zbl 0469.30024
[20] V. P. Gurarii, On Levinson’s theorem concerning families of analytic functions, Seminars in Math., V. A. Steklov Math. Inst., Leningrad, 19 (1972), 124–127.
[21] L. Hörmander, The Analysis of Linear Partial Differential Operators I, II, Springer-Verlag, Berlin, 1983.
[22] R. Hornblower, A growth condition for the MacLane class A, Proc. London Math. Soc. 23 (1971), 371–394. · Zbl 0223.30043
[23] A. Jonsson and H. Wallin, A Whitney extension theorem in L p and Besov spaces, Ann. Inst. Fourier 28 (1978), 139–192. · Zbl 0335.46024
[24] B. I. Korenblum, Invariant subspaces of the shift operator on a weighted Hilbert space, Math. USSR Sb. 89 (1972), 110–137.
[25] A. M. Kotochigov, Interpolation by analytic functions smooth up to the boundary, J. Soviet Math. 4 (1975), 4. · Zbl 0341.41006
[26] N. Levinson, Gap and Density Theorems, AMS, Providence, 1940. · JFM 66.0332.01
[27] V. I. Macaev, Ph.D. Thesis, Harkov, 1964.
[28] G. R. MacLane, Asymptotic values of holomorphic functions, Rice Univ. Studies 49, No. 1, 1963. · Zbl 0121.30203
[29] S. Mandelbrojt, Series de Fourier et classes quasi analytiques de fonctions, Gauthier-Villars, Paris, 1935. · Zbl 0013.11006
[30] S. Mandelbrojt, Series adherentes, Paris, 1952.
[31] L. Nirenberg, A proof of the Malgrange preparation theorem, Springer Lecture Notes in Math. 192 (1971), 97–105. · Zbl 0212.10702
[32] N. A. Shirokov, Free interpolation in spaces $C_{n,ega} {A}$, Math. Sbornik 117 (1982), 337–358.
[33] N. A. Shirokov, Analytic functions smooth up to the boundary, Springer Lecture Notes in Math. 1312 (1988). · Zbl 0656.30029
[34] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton, 1970. · Zbl 0207.13501
[35] A. L. Volberg and B. Jöricke, The summability of logarithm of an almost analytic function and a generalization of a theorem of Levinson and Cartwright, Math. USSR Sb. 130 (1986), 335–348.
[36] A. L. Volberg and S. V. Konyagin, On measures with the doubling condition, Math. USSR Izvestiya 30 (1988), 629–638. · Zbl 0727.28012
[37] A. L. Volberg, The Lojasiewicz inequality for very smooth functions, LOMI preprint, E-15-88, Leningrad, 1989, 54pp.
[38] S. Warschawski, On conformai mapping of infinite strips, Trans. Am. Math. Soc. 51 (1942), 280–335. · Zbl 0028.40303
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