Intrepolation problems in Hörmander algebras

Myriam Ounaies

May 8, 2009

Abstract: We call Hörmander algebras the spaces A_{p}(ℂ) of entire functions f such that, for all z in ℂ,

|f(z)| ≤ Ae^{Bp(z)},

where A and B are some positive constants (depending on f) and p is a subharmonic weight. We consider the following interpolation problem: Given a discrete sequence {a_{j}} of complex numbers and a sequence of complex values {b_{j}}, under what conditions does there exist a function f ∈ A_{p}(ℂ) such that f(a_{j}) = b_{j} for all j ? In other words, what is the trace of A_{p}(ℂ) on {a_{j}}? We say that {a_{j}} is an interpolating sequence if the trace is defined by the space of all {b_{j}} satisfying |b_{j}| ≤ A′e^{B′p(a_{j})}, for some constants A′, B′ > 0. We use Hormander’s L^{2}-estimates for the ¯∂-equation to describe the trace when the weight p is radial and doubling and to characterize the interpolating sequences for more general weights.