Fatou theorem

2020 Mathematics Subject Classification: Primary: 31A20 [MSN][ZBL] $ \newcommand{\abs}[1]{\left|#1\right|} $

The Fatou theorem is a theorem in the theory of functions of a complex variable: Suppose that the harmonic function $u(z)$, $z=r\mathrm{e}^{\mathrm{i}\phi}$, can be represented in the unit disc $U=\{ z\in\C : \abs{z} < 1 \}$ by a Poisson–Stieltjes integral \[ u(z) = \int \frac{1-r^2}{1-2r\cos(\theta-\phi)+r^2} \rd \mu(\zeta), \quad \zeta = \mathrm{e}^{\mathrm{i}\theta}, \] where $\mu$ is a Borel measure concentrated on the unit circle $T=\{ z\in\C : \abs{z} = 1 \}$, $\int\rd\mu(\xi)=1$. Then almost-everywhere with respect to the Lebesgue measure on $T$, $u(z)$ has angular boundary values.

This Fatou theorem can be generalized to harmonic functions $u(x)$, $x\in\R^n$, $n\geq2$, that can be represented by a Poisson–Stieltjes integral in Lyapunov domains $D\subset\R^n$ (see [REF], [REF]). For Fatou's theorem for radial boundary values of multiharmonic functions in the polydisc \[ U^n = \left\{ z=(z_1,\ldots,z_n)\in\C^n : \abs{z_j}<1, j=1,\ldots,n \right\} \] see [REF], [REF].

If $f(z)$ is a bounded analytic function in $U$, then almost-everywhere with respect to the Lebesgue measure on $T$ it has angular boundary values.

This Fatou theorem can be generalized to functions of bounded characteristic (see [REF]). Points $\zeta$ at which there is an angular boundary value $f(\zeta)$ are called Fatou points. Regarding generalizations of the Fatou theorem for analytic functions $f(z)$ of several complex variables $z=(z_1,\ldots,z_n)$, $n\geq 2$, see [REF]; it turns out that for $n\geq 2$ there are also boundary values along complex tangent directions.

If the coefficients of a power series $\sum_{k=0}^\infty a_k z^k$ with unit disc of convergence $U$ tend to zero, $\lim_{k\rightarrow\infty a_k=0}$, then this series converges uniformly on every arc $\alpha\leq\theta\leq\beta$ of the circle $T$ consisting only of regular boundary points for the sum of the series.

If $\lim_{k\rightarrow\infty} a_k=0$ and the series converges uniformly on an arc $\alpha\leq\theta\leq\beta$, 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 [Fa].

References

[Fa]	P. Fatou, "Séries trigonométriques et séries de Taylor" Acta Math., 30 (1906) pp. 335–400
[KhCh]	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
[Pr]	I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen", Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[PrKu]	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)
[Ru]	W. Rudin, "Function theory in polydiscs", Benjamin (1969)
[So]	E.D. Solomentsev, "On boundary values of subharmonic functions" Czechoslovak. Math. J., 8 (1958) pp. 520–536 (In Russian) (French summary)
[Zy]	A. Zygmund, "Trigonometric series", 1–2, Cambridge Univ. Press (1988)

Comments

For Lyapunov domain see Lyapunov surfaces and curves. For Fatou theorems in $\C^n$ see [Ru2], [St], [NaSt].

References

[Ho]	K. Hoffman, "Banach spaces of analytic functions", Prentice-Hall (1962)
[La]	E. Landau, "Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie", Das Kontinuum und andere Monographien, Chelsea, reprint (1973)
[NaSt]	A. Nagel, E.M. Stein, "On certain maximal functions and approach regions" Adv. in Math., 54 (1984) pp. 83–106
[Ru2]	W. Rudin, "Function theory in the unit ball in $\C^n$", Springer (1980)
[St]	E.M. Stein, "Boundary behavior of holomorphic functions of several complex variables", Princeton Univ. Press (1972)