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Embeddings of rearrangement invariant spaces that are not strictly singular. (English) Zbl 0987.46032

A map between two normed spaces is said strictly singular if there does not exist an infinite dimensional subspace of the domain on which the operator is an isomorphism. The question when the natural embedding are strictly singular seems to be difficult problem. For this reason, they study the question. Given a rearrangement invariant space \(E\) on \([0,1]\), when is the natural embedding \(E\subset L_1([0,1])\) strictly singular?
Though they are not able to give complete answer to this question, they give partial answer to the following conjecture: the natural embedding of a rearrangement invariant space \(E\) into on \(L_1([0,1])\) is strictly singular if and only if \(G\) does not embed into \(E\) continuously, where \(G\) is the closure of the simple functions in the Orlicz space \(L_\Phi\), with \(\Phi(x)=\exp(x^2)-1\).
The best result they have obtained regarding this conjecture is the following. If \(E\) is \(D\)-convex this conjecture is true. A rearrangement invariant space \(E\) is \(D\)-convex if there is a family of Orlicz functions \(\Phi_\alpha: {\mathbb R}\in [0,\infty)\), and a constant \(c>0\), such that \(c^{-1}\|x\|_E\leq\sup_\alpha \|x\|_{\Phi_\alpha}\leq c\|x\|_E\). From here they are able to get a weaker version of this conjecture. If the natural embedding \(E\subset L_1\) is not strictly singular, then \(G_1\) embeds into \(E\) continuously. By \(G_1\) denoting Lorentz space \(\Lambda(\varphi)\) with \(\varphi(t)=2 / \sqrt{\log(e^2/t)}\).

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B38 Linear operators on function spaces (general)
60G50 Sums of independent random variables; random walks
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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