The Generalized Abel Equations for Schwartz Distributions

Abstract

The class of equations $aI^\alpha f + bK^\alpha f = g$ is analyzed for data g and solutions f belonging to the space $\mathcal {D}'(\mathcal {R})$ of Schwartz distributions. f has compact support contained in $[ - 1,1]$, and g is known only on $( - 1,1)$. $I^\alpha $ and $K^\alpha $ are the usual operators of fractional integration defined as convolutions. The equation is to be satisfied as an identity between distributions on the open interval $( - 1,1)$. The coefficients a and b are infinitely differentiable functions on $( - 1,1)$ subject to certain growth conditions at the endpoints.It is shown that in this setting the equation $aI^\alpha f + bK^\alpha f = g$ is equivalent to a boundary value problem for functions analytic off the real axis. The class of analytic functions furnishing the solutions is characterized in terms of its growth rate at infinity and its limiting behavior at the real axis. Solutions to the generalized Abel equations are found explicitly for arbitrary distribution data and all c...