Fatou theorem

in the theory of functions of a complex variable

Suppose that the harmonic function , , can be represented in the unit disc by a Poisson–Stieltjes integral

where is a Borel measure concentrated on the unit circle , . Then almost-everywhere with respect to the Lebesgue measure on , has angular boundary values (cf. Angular boundary value).

This Fatou theorem can be generalized to harmonic functions , , , that can be represented by a Poisson–Stieltjes integral in Lyapunov domains (see , ). For Fatou's theorem for radial boundary values (cf. Radial boundary value) of multiharmonic functions in the polydisc see , .

If is a bounded analytic function in , then almost-everywhere with respect to the Lebesgue measure on it has angular boundary values.

This Fatou theorem can be generalized to functions of bounded characteristic (cf. Function of bounded characteristic) (see ). Points at which there is an angular boundary value are called Fatou points. Regarding generalizations of the Fatou theorem for analytic functions of several complex variables , , see ; it turns out that for there are also boundary values along complex tangent directions.

If the coefficients of a power series with unit disc of convergence tend to zero, , then this series converges uniformly on every arc of the circle consisting only of regular boundary points for the sum of the series.

If and the series converges uniformly on an arc , it does not follow that the points of this arc are regular for the sum of the series.

Theorems 1), 2) and 3) were proved by P. Fatou [1].

References

[1]	P. Fatou, "Séries trigonométriques et séries de Taylor" Acta Math. , 30 (1906) pp. 335–400
[2]	I.I. Privalov, P.I. Kuznetsov, "On boundary problems and various classes of harmonic and subharmonic functions on an arbitrary domain" Mat. Sb. , 6 : 3 (1939) pp. 345–376 (In Russian) (French summary)
[3]	E.D. Solomentsev, "On boundary values of subharmonic functions" Czechoslovak. Math. J. , 8 (1958) pp. 520–536 (In Russian) (French summary)
[4]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
[5]	W. Rudin, "Function theory in polydiscs" , Benjamin (1969)
[6]	I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[7]	G.M. Khenkin, E.M. Chirka, "Boundary properties of holomorphic functions of several complex variables" J. Soviet Math. , 5 : 5 (1976) pp. 612–687 Itogi Nauk. i Tekhn. Sovr. Probl. Mat. , 4 (1975) pp. 13–142

Comments

For Lyapunov domain see Lyapunov surfaces and curves. For Fatou theorems in see [a3]–[a5].

References

[a1]	E. Landau, "Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie" , Das Kontinuum und andere Monographien , Chelsea, reprint (1973)
[a2]	K. Hoffman, "Banach spaces of analytic functions" , Prentice-Hall (1962)
[a3]	W. Rudin, "Function theory in the unit ball in " , Springer (1980)
[a4]	E.M. Stein, "Boundary behavior of holomorphic functions of several complex variables" , Princeton Univ. Press (1972)
[a5]	A. Nagel, E.M. Stein, "On certain maximal functions and approach regions" Adv. in Math. , 54 (1984) pp. 83–106