Leon Ehrenpreis

Abstract:   Denote by O⁺ and O^­ the positive and negative orthants in Rⁿ. The wedges W⁺ and W^­ are the tubes over O⁺ and O^­. The edge is the intersection of the wedges, which is the imaginary space. Let f⁺ and f^­ be functions which are holomorphic on W⁺ and W^­. If f⁺ = f^­ on the edge then there is a function F which is holomorphic on all of Cⁿ which extends f⁺ and f^­.

We shall sharpen and extend this result in several directions:

1. Holomorphicity is defined by the Cauchy Riemann operator (d bar) which we shall replace by a general class of differential operators P(D).
2. Problems in the spirit of Bill Clinton: What is "is" (for us "is" means =)?
   1. What is the weakest sense in which f⁺ = f^­ on the edge? (Problem of Norman Levinson)
   2. (Problem of Fefferman-Nirenberg-Paul Yang) Instead of Pf ^± = 0 we require only that Pf ^± vanish rapidly at the edge.
   3. (quasianalyticity problem) Instead of Pf ^± = 0 we give bounds on the P^mf^±.