Best one-sided L1-approximation by harmonic functions

Let f is an element of C((B) over bar), where B is the open unit ball in R-n (n greater than or equal to 2), and let U-B(f) denote the collection of functions h in C((B) over bar) which are harmonic on B and satisfy h less than or equal to f on B. A function h* in U-B(f) is called a best harmonic one-sided L-1-approximant to f if integral((B) over bar)\f - h*\ less than or equal to integral\ f - h\ for all h in U-B(f). This paper characterizes such approximants and discusses questions of existence and uniqueness. Corresponding results for approximation on the cylinder B x R are also established, but the proofs in this case are more difficult and rely on recent work concerning tangential harmonic approximation. The characterizations are quite different in nature from those recently obtained for harmonic L-1-approximation without a one-sidedness condition.