### A note on a paper by Andreotti and Hill concerning the Hans Lewy problem

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We consider the topological algebra of (Taylor) multipliers on spaces of real analytic functions of one variable, i.e., maps for which monomials are eigenvectors. We describe multiplicative functionals and algebra homomorphisms on that algebra as well as idempotents in it. We show that it is never a Q-algebra and never locally m-convex. In particular, we show that Taylor multiplier sequences cease to be so after most permutations.

The theory of Mellin analytic functionals with unbounded carrier is developed. The generalized Mellin transform for such functionals is defined and applied to solve the Laplace-Beltrami type singular equations on a hyperbolic space. Then the asymptotic expansion of solutions is found.

We develop an elementary theory of Fourier and Laplace transformations for exponentially decreasing hyperfunctions. Since any hyperfunction can be extended to an exponentially decreasing hyperfunction, this provides simple notions of asymptotic Fourier and Laplace transformations for hyperfunctions, improving the existing models. This is used to prove criteria for the uniqueness and solvability of the abstract Cauchy problem in Fréchet spaces.

We prove precise decomposition results and logarithmically convex estimates in certain weighted spaces of holomorphic germs near ℝ. These imply that the spaces have a basis and are tamely isomorphic to the dual of a power series space of finite type which can be calculated in many situations. Our results apply to the Gelfand-Shilov spaces $S{\xb9}_{\alpha}$ and $S{\u2081}^{\alpha}$ for α > 0 and to the spaces of Fourier hyperfunctions and of modified Fourier hyperfunctions.

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

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 $\tilde{}={\u2102}^{n+1}$ 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.