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A comparison between the closed modular ideals in \(\ell ^ 1(w)\) and \(L^ 1(w)\). (English) Zbl 0654.46056
Let \(\omega\) be a submultiplicative weight on \(R^+\) and let \(\ell^ 1(\omega)\) and \(L^ 1(\omega)\) be the associated Beurling algebras on N and \(R^+\). The author establishes the existence of a bijection \(\theta\) from a large class of closed modular ideals in \(L^ 1(\omega)\) and a corresponding class of closed ideals in \(\ell^ 1(\omega)\). He exhibits an isomorphism between the annihilator of an ideal I in \(L^ 1(\omega)\) and the annihilator of \(\theta\) (I). Of interest in themselves are explicit formulas for an analytic extension of the Laplace transform, and for the Borel transform, of a function in the annihilator of one of these ideals.
Reviewer: H.Hedenmalm

46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46A45 Sequence spaces (including Köthe sequence spaces)
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