Let $\mu \in {\left({ℝ}^d\right)}^{\text{'}}$ be an analytic functional and let $T_{\mu }$ be the corresponding convolution operator on Sato’s space $\left({ℝ}^d\right)$ of hyperfunctions. We show that $T_{\mu }$ is surjective iff $T_{\mu }$ admits an elementary solution in $\left({ℝ}^d\right)$ iff the Fourier transform μ̂ satisfies Kawai’s slowly decreasing condition (S). We also show that there are $0\ne \mu \in {\left({ℝ}^d\right)}^{\text{'}}$ such that $T_{\mu }$ is not surjective on $\left({ℝ}^d\right)$.

Let A(Ω) denote the real analytic functions defined on an open set Ω ⊂ ℝⁿ. We show that a partial differential operator P(D) with constant coefficients is surjective on A(Ω) if and only if for any relatively compact open ω ⊂ Ω, P(D) admits (shifted) hyperfunction elementary solutions on Ω which are real analytic on ω (and if the equation P(D)f = g, g ∈ A(Ω), may be solved on ω). The latter condition is redundant if the elementary solutions are defined on conv(Ω). This extends and improves previous...

Let L(z) be the Lie norm on $\tilde{}={ℂ}^{n+1}$ and L*(z) the dual Lie norm. We denote by ${}_{\Delta }\left(\tilde{B}\left(R\right)\right)$ the space of complex harmonic functions on the open Lie ball $\tilde{B}\left(R\right)$ and by $Ex{p}_{\Delta }(\tilde{};(A,L*))$ the space of entire harmonic functions of exponential type (A,L*). A continuous linear functional on these spaces will be called a harmonic functional or an entire harmonic functional. We shall study the conical Fourier-Borel transformations on the spaces of harmonic functionals or entire harmonic functionals.

For an analytic functional $S$ on ${ℂ}^n$, we study the homogeneous convolution equation S * f = 0 with the holomorphic function f defined on an open set in ${ℂ}^n$. We determine the directions in which every solution can be continued analytically, by using the characteristic set.

We determine the convolution operators $T_{\mu }:=\mu *$ on the real analytic functions in one variable which admit a continuous linear right inverse. The characterization is given by means of a slowly decreasing condition of Ehrenpreis type and a restriction of hyperbolic type on the location of zeros of the Fourier transform μ̂(z).