Parameter dependence of solutions of the Cauchy–Riemann equation on weighted spaces of smooth functions

Abstract

Let Ω be an open subset of 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 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.

The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes

“…Our main goal is to derive sufficient conditions on V and ( n ) n∈ℕ such that is surjective. We recall the main result of [22], which sets the course of the present paper. Here EV( ) ∶= EV( , ℂ) and EV ( ) is the kernel of in EV( ) , i.e.…”

“…The parameter dependence problem for a variety of partial differential operators on several spaces of (generalised) differentiable functions has been extensively studied, see e.g. [4,6,7,16,31,32] and the references and background in [3,22]. The answer to this problem for the Cauchy-Riemann operator is affirmative since the Cauchy-Riemann operator on the space C ∞ ( , E) of E-valued smooth functions is surjective if E = C ∞ (U) ( O(U) , D(V) � ) by [8, Corollary 3.9, p. 1112] which is a consequence of the splitting theory of Bonet and Domański for PLS-spaces [3,4], the topological isomorphy of C ∞ ( , E) to Schwartz' -product C ∞ ( ) E and the fact that ∶ C ∞ ( ) → C ∞ ( ) is surjective on the nuclear Fréchet space C ∞ ( ) (with its usual topology).…”

“…The first and the last result cover the case that E = C ∞ (U) or O(U) whereas the last covers the case E = D(V) � as well. More examples of the second or third kind of such spaces E are arbitrary Fréchet-Schwartz spaces, the space S(ℝ d ) � of tempered distributions, the space D(V) � of distributions, the space D (w) (V) � of ultradistributions of Beurling type and some more (see [4, 8, Corollary 4.8, p. 1116] and [22,Example 3,p. 7]).…”

“…Looking at Theorem 1, the main obstacle is to prove that E (a n ), (ℂ ⧵ K) satsifies property ( ) . In [22,Corollary 14,p. 18] this is accomplished for K = ∅ and sequences (a n ) n∈ℕ such that a n ↗ 0 or a n ↗ ∞ .…”

The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes

“…The parameter dependence problem for a variety of partial differential operators on several spaces of (generalised) differentiable functions has been extensively studied, see e.g. [4,6,7,37,38,18] and the references and background in [3,26]. The answer to this problem for the Cauchy-Riemann operator is affirmative since the Cauchy-Riemann operator…”