Let and , where . If is continuous and satisfies , then the Poisson integral of yields a harmonic function on with boundary values on . It is known that, for functions which fail to satisfy this integrability condition, the corresponding Dirichlet problem can be solved by suitably modifying the Poisson kernel. Although these solutions are certainly not unique, the paper under review presents a type of uniqueness theorem which answers a question posed by D. Siegel.
To be more precise, suppose that for some in . Then there exists a harmonic function on with boundary values such that as , where denotes the mean value of with respect to surface area measure on the hemisphere . Further, all other solutions satisfying this growth condition are of the form , where is a polynomial in of degree at most and even with respect to the variable .