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On the boundary values of harmonic functions in R 3

The purpose of this paper is to exhibit three theorems about the boundary values of harmonic functions, defined in R a regions which are bounded by Liapunov surfaces. Theorem 1 shows the existence almost everywhere of nontangential boundary values of positive harmonic functions. A full proof of this theorem is given. Theorem 2 assures the existence almost everywhere of nontangential boundary values for functions bounded in cones with vertex on the boundary and lying in the region. Theorem 3, finally, gives a necessary and sufficient condition for the existence of non-tangential boundary values, originally derived by Marcinkiewicz and Zygmund and later generalized by Stein. As the proofs of the two latter theorems differ from proofs published elsewhere only in the technical aspect, these are not included here.