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On spectral synthesis in some Hardy-Sobolev spaces. (English) Zbl 0794.46023

The Hardy-Sobolev spaces to be considered are the potential spaces \(I_ s H^ 1=\{f: f=I_ s *h, h\in H^ 1\}\), \(0< s\leq d\), where \(I_ s\) is the Riesz potential of order \(s\), that is, \(I_ s(x)= | x|^{s- d}\) \((=\log| x|\) if \(s=d)\) and \(H^ 1\) is the usual Fefferman- Stein Hardy space. We endow, \(I_ sH^ 1\) with the Banach space norm \(\| f\|= \| h\|_{H^ 1}\), \(f= I_ s* h\). In this paper, we give a necessary and sufficient condition so that a function \(f\) can be approximated in \(I_ s H^ 1\), \(0< s<2\), by functions in \(C^ \infty_ 0(F^ c)\), where \(F\) is a closed set of \(\mathbb{R}^ d\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
43A45 Spectral synthesis on groups, semigroups, etc.
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