Vol. 90, No. 2, 1980

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Vector-valued distributions having a smooth convolution inverse

Hector O. Fattorini

Vol. 90 (1980), No. 2, 347–372

Let E, X be complex Banach spaces, (E,X) the space of linear operators from E into X equipped with its usual norm. We denote by 𝒟′(E) the space of E-valued distributions defined in −∞ < t < and by 𝒟′0(E) the subspace thereof consisting of distributions with support in t 0. A distribution P ∈𝒟′0((X;E)) is said to have a convolution inverse (in symbols, P ∈𝒟′0((E;X))1 or simply P 𝒟′01) if there exists S ∈𝒟′0((E;X)) such that

P ∗S = δ⊗ I,  S ∗P = δ⊗ J

where δ is the Dirac measure and I (resp. J) denotes the identity operator in E (resp. X). We examine the problem of characterizing those P which possess a convolution inverse S = P1 being smooth in various senses: infinitely differentiable, in a quasi-analytic class, analytic, etc.

Mathematical Subject Classification 2000
Primary: 35E05
Received: 15 June 1979
Revised: 17 October 1979
Published: 1 October 1980
Hector O. Fattorini