The converse of the Fatou theorem for positive harmonic functions

Abstract

The converses of these theorems are general not true. If v(z) is positive however, both converses can be proved. One result is that if v(rei6) is a bounded function harmonic IzI <1, and if its boundary function v(O) is defined as the limit, wherever it exists, of v(z) as z-*e@6 in angle, then v(O) is a summable function which is precisely equal to the derivative of its indefinite integral. The converse of Theorem A for positive functions follows readily from known theorems, and it is the main object of this paper to deduce from it a strengthened form of the converse of Theorem B for positive functions. We shall have occasion to use the theorem(') that a harmionic function has the representation (1) if and only if it can be written as the difference of two non-negative (or two positive) harmonic functions. In particular, every positive harmonic function has the representation (1) with V(O) increasing., 2. The converse of Theorem A for positive functions. It will be simplest to infer the converse of Theorem A for positive functions from a series of remarks. (i) The limit (if it exists) V(1)(O) =limt,o [ V(O+t) V(O-t) ]/2t is known as the generalized symmetric derivative of V(O).