On the existence of boundary values of a class of Beppo Levi functions

Abstract

where I PI is the distance from the origin to a point P in Rm. Beurling [1, p. 5] has proved that if f is harmonic in U and m = 2 and oc = 0 then f has finite radial limits except when approaching a certain exceptional set on the boundary of U having logarithmic capacity zero. After that Deny [5, p. 175] proved, for a general m and a= 0, that if f is a continuous Beppo Levi function then f has finite radial limits except on an exceptional set having capacity zero with respect to the kernel r (m 2). For m = 2, Carleson [4, p. 48] proved a corresponding theorem for a general ac and an exceptional set having capacity zero with respect to the kernel r. Using techniques from [6] and [10] and a method different from that of Carleson we shall generalize to higher dimensions his theorem with a general ac and with J absolutely continuous in the same way as a Beppo Levi function but for a half-space R+ bounded by a hyperplane instead of for the sphere (Theorem 1). Let P = (X,y) = (x1,x2, .*,xmlI,y) be a point in R' and y > 0 the distance from P to the boundary of R+m which we identify with the (m 1)-dimensional Euclidean space R1 with points X (x1, x2, .., xm_ 1). Then the analogue of (1.1) is the condition