Helmholtz solutions for the fractional Laplacian and other related operators

Abstract

We show that the bounded solutions to the fractional Helmholtz equation, [Formula: see text] for [Formula: see text] in [Formula: see text], are given by the bounded solutions to the classical Helmholtz equation [Formula: see text] in [Formula: see text] for [Formula: see text] when [Formula: see text] is additionally assumed to be vanishing at [Formula: see text]. When [Formula: see text], we show that the bounded fractional Helmholtz solutions are again given by the classical solutions [Formula: see text]. We show that this classification of fractional Helmholtz solutions extends for [Formula: see text] and [Formula: see text] when [Formula: see text]. Finally, we prove that the classical solutions are the unique bounded solutions to the more general equation [Formula: see text] in [Formula: see text], when [Formula: see text] is complete Bernstein and certain regularity conditions are imposed on the associated weight [Formula: see text].