Luzin-Privalov theorems

2020 Mathematics Subject Classification: Primary: 30D40 [MSN][ZBL]

The Luzin–Privalov theorems in the theory of functions of a complex variable are classical results of N.N. Luzin and I.I. Privalov that clarify the character of a boundary uniqueness property of analytic functions (cf. Uniqueness properties of analytic functions) (see [LuPr]).

1) Let $f(z)$ be a meromorphic function of the complex variable $z$ in a simply-connected domain $D$ with rectifiable boundary $\Gamma$. If $f(z)$ takes angular boundary values zero on a set $E\subset\Gamma$ of positive Lebesgue measure on $\Gamma$, then $f(z)=0$ in $D$. There is no function meromorphic in $D$ that has infinite angular boundary values on a set $E\subset\Gamma$ of positive measure.

2) Let $w=f(z)$ be a meromorphic function in the unit disc $D=\{z:\left|z\right|<1\}$ other than a constant and having radial boundary values (finite or infinite) on a set $E$ situated on an arc $\sigma$ of the unit circle $\Gamma=\{z:\left|z\right|=1\}$ that is metrically dense and of the second Baire category (cf. Baire classes) on $\sigma$. Then the set $W$ of its radial boundary values on $E$ contains at least two distinct points. Metric density of $E$ on $\sigma$ means that every portion of $E$ on $\sigma$ has positive measure. This implies that if the radial boundary values of $f(z)$ on a set $E$ of the given type are equal to zero, then $f(z)=0$ in $D$. Moreover, there is no meromorphic function in the unit disc that takes infinite radial boundary values on a set $E$ of the given type.

Luzin and Privalov (see [LuPr], [Pr]) constructed examples to show that neither metric density nor second Baire category are by themselves sufficient for the assertion in 2 to hold.

See also Boundary properties of analytic functions; Luzin examples; Cluster set; Privalov theorem; Riesz theorem.

References

[CoLo]	E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets", Cambridge Univ. Press (1966) pp. Chapt. 9
[Lo]	A. Lohwater, "The boundary behaviour of analytic functions" Itogi Nauk. i Tekhn. Mat. Anal., 10 (1973) pp. 99–259 (In Russian)
[LuPr]	N.N. [N.N. Luzin] Lusin, I.I. [I.I. Privalov] Priwaloff, "Sur l'unicité et la multiplicité des fonctions analytiques" Ann. Sci. Ecole Norm. Sup. (3), 42 (1925) pp. 143–191
[Pr]	I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen", Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)