Continuous Potentials and Linear Mass Distributions

Abstract

is continuous on the entire plane. The requirement that y possess a continuous tangent can, of course, be weakened to allow the tangent to be piecewise continuous. On the other hand, it is known that the smoothness condition cannot be discarded altogether. The present paper stems from an attempt to determine the extent to which the conditions on y can be weakened without destroying continuity of the potential. As it turns out-and this is the point we wish to emphasize-not only is smoothness of the curve an excessively stringent requirement, but in fact the initial assumption that the mass is distributed one-dimensionally is only incidentally connected with continuity of its potential. One of our main objectives here is to exhibit a simple criterion for continuity of a potential at a point. Although this criterion does not seem to have been discussed in the literature, it is closely related to previous work on normal families of subharmonic functions [1]. The derivation makes use of the mean-value operator, the importance of which was largely unrecognized in the early development of potential theory. It follows from the continuity criterion that a general Lipschitz condition on the mass distribution will suffice to ensure continuity of the potential. This in turn leads at once to the classical results on continuity of line and area distributions.