DOI: 10.1515/gmj-2026-3044 ISSN: 1072-947X

A completeness theorem on the boundary in Besov spaces

Alberto Cialdea, Carmine Sebastiano Mare, David Natroshvili
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Abstract

Let

{ ω k } {\{\omega_{k}\}}
be a basis of polynomial solutions to the equation
E ⁢ u = 0 {Eu=0}
, where E is a higher-order elliptic operator with real constant coefficients and no lower-order terms, in any number of variables. The aim of the paper is to prove that the system
{ ( ω k , ∂ ν ⁡ ω k , … , ∂ ν m - 1 ⁡ ω k ) } {\{(\omega_{k},\partial_{\nu}\omega_{k},\dots,\partial^{m-1}_{\nu}\omega_{k})\}}
is complete in the Besov space
[ B p , p 1 - 1 / p ⁢ ( ∂ ⁡ Ω ) ] m {[B^{1-{1}/{p}}_{p,p}(\partial\Omega)]^{m}}
( 1 ≤ p < ∞ ) {(1\leq p<\infty)}
, Ω being a
C ∞ {C^{\infty}}
bounded domain in
ℝ n {\mathbb{R}^{n}}
such that
ℝ n ∖ Ω ¯ {\mathbb{R}^{n}\setminus\overline{\Omega}}
is connected.

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