Abstract
Let varOmega be an open subset of mathbb {R}^{2} and E a complete complex locally convex Hausdorff space. The purpose of this paper is to find conditions on certain weighted Fréchet spaces mathcal {EV}(varOmega ) of smooth functions and on the space E to ensure that the vector-valued Cauchy–Riemann operator {overline{partial }}:mathcal {EV}(varOmega ,E)rightarrow mathcal {EV}(varOmega ,E) is surjective. This is done via splitting theory and positive results can be interpreted as parameter dependence of solutions of the Cauchy–Riemann operator.
Highlights
The purpose of this paper is to find conditions on certain weighted Fréchet spaces EV(Ω) of smooth functions and on the space E to ensure that the vector-valued Cauchy–Riemann operator ∂ : EV(Ω, E) → EV(Ω, E) is surjective
This is done via splitting theory and positive results can be interpreted as parameter dependence of solutions of the Cauchy– Riemann operator
Let E be a linear space of functions on a set U and P(∂) : F(Ω) → F(Ω) be a linear partial differential operator with constant coefficients which acts continuously on a locally convex Hausdorff space of differentiable scalar-valued functions F(Ω) on an open set Ω ⊂ Rn
Summary
Let E be a linear space of functions on a set U and P(∂) : F(Ω) → F(Ω) be a linear partial differential operator with constant coefficients which acts continuously on a locally convex Hausdorff space of (generalized) differentiable scalar-valued functions F(Ω) on an open set Ω ⊂ Rn. The necessary condition of surjectivity of the partial differential operator P(∂) was studied in many papers, e.g. in [1,23,28,48,67] on C∞-smooth functions and distributions, in [9,26,43] on real analytic functions, in [8,14] on Gevrey classes, in [10,12,41,42,55] on ultradifferentiable functions of Roumieu type, in [22] on ultradistributions of Beurling type, in [7,11] on ultradifferentiable functions and ultradistributions and in [47] on the multiplier space OM. We close this section with a special case of our main theorem where (Ωn)n∈N is a sequence of strips along the real axis (see Corollary 17) and for example νn(z) := exp(an| Re(z)|γ ) for some 0 < γ ≤ 1 and an 0 (see Corollary 18)
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