Polynomial Hulls and H1 Control for a Hypoconvex Constraint

We say that a subset of C n is hypoconvex if its complement is the union of complex hyperplanes. Let be the closed unit disk in C , ? = @. We prove two conjectures of Helton and Marshall. Let be a smooth function on ? C n whose sublevel sets have compact hypoconvex bers over ?. Then, with some restrictions on , if Y is the set where is less than or equal to 1, the polynomial convex hull of Y is the union of graphs of analytic vector valued functions with boundary in Y. Furthermore, we show that the innmum inf f2H 1 (() n k(z; f(z))k 1 is attained by a unique bounded analytic f which in fact is also smooth on ?. We also prove that if varies smoothly with respect to a parameter, so does the unique f just found. We address two conjectures of Helton and Marshall from HMa, p. 183] which generalize previous theorems regarding an H 1 control problem over the disk and polynomial hulls of compact sets in C n+1 bered over the circle in C. If Y is a compact set in C n , then the polynomial (convex) hull b Y of Y is given by b Y = fz 2 C n jP(z)j sup w2Y jP(w)j for all polynomials P on C n g.