Difference between revisions of "Luzin-Privalov theorems"

in the theory of functions of a complex variable

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 [1]).

1) Let be a meromorphic function of the complex variable in a simply-connected domain with rectifiable boundary . If takes angular boundary values zero on a set of positive Lebesgue measure on , then in . There is no function meromorphic in that has infinite angular boundary values on a set of positive measure.

2) Let be a meromorphic function in the unit disc other than a constant and having radial boundary values (finite or infinite) on a set situated on an arc of the unit circle that is metrically dense and of the second Baire category (cf. Baire classes) on . Then the set of its radial boundary values on contains at least two distinct points. Metric density of on means that every portion of on has positive measure. This implies that if the radial boundary values of on a set of the given type are equal to zero, then in . Moreover, there is no meromorphic function in the unit disc that takes infinite radial boundary values on a set of the given type.

Luzin and Privalov (see [1], [2]) constructed examples to show that neither metric density nor second Baire category are by themselves sufficient for the assertion in 2 to hold.

References

 [1] 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 [2] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) [3] A. Lohwater, "The boundary behaviour of analytic functions" Itogi Nauk. i Tekhn. Mat. Anal. , 10 (1973) pp. 99–259 (In Russian)