On generalized Lorentz-Zygmund spaces. (English) Zbl 0956.46020

Let \((\mathcal R,\mu)\) be a suitable measure space. Given a function \(f\) on \(\mathcal R\) its non-decreasing rearrangement \(f^*\) is defined by \(f^*=\inf\{\lambda>0:\mu(\{x\in\mathcal R:|f(x)|>\lambda\})\leq t\}\) for \(t\geq 0\). The so called Lorentz-Zygmund spaces \(L_{p,q;\alpha}\), \(0<p,q\leq\infty\), \(\alpha\in\mathbb R\), were introduced by C. Bennett and K. Rudnick [Diss. Math. 175, 67 p. (1980; Zbl 0456.46028)] as the set of functions \(f\) which satisfy \(\|t^{1/p-1/q}(1+|\log t|)^\alpha f^*(t)\|_{q,(0,\mu(\mathcal R))}<\infty\). The class of Lorentz-Zygmund spaces contains Lebesgue spaces, Lorentz spaces and Zygmund classes and further important spaces. Recently, D. E. Edmunds, P. Gurka and B. Opic in [Indiana Univ. Math. J. 44, No. 1, 19-43 (1995; Zbl 0826.47021)] studied the double-exponential integrability of convolution operators and extended the theory of Lorentz-Zygmund spaces to the so called generalized Lorentz-Zygmund (GLZ) spaces by introducing a second trier of logarithms. This structure was then intensively studied by D. E. Edmunds, P. Gurka, B. Opic, L. Pick, W. Trebels and other authors.
The present paper is devoted to a detailed study of GLZ spaces \(L_{p,q;\mathbb A,\mathbb B}\) and \(L_{(p,q;\mathbb A,\mathbb B)}\) consisting of all \(\mu\)-measurable functions \(f\) on \(\mathcal R\) such that the norms \((\int_0^{\mu(\mathcal R)}t^{1/p-1/q}\ell^{\mathbb A}(t)\ell\ell^{\mathbb B}(t) f^*(t)d\mu)^{1/q}\) and \((\int_0^{\mu(\mathcal R)} t^{1/p-1/q}\ell^{\mathbb A}(t)\ell\ell^{\mathbb B}(t)f^{**}(t))^{1/q}\) are finite, respectively. Here \(f^{**}(t)=t^{-1}\int_0^t f^*(s)ds\), \(\mathbb A=(\alpha_0,\alpha_\infty),\mathbb B=(\beta_0,\beta_\infty)\in\mathbb R^2\), \(\ell(t)=1+|\log t|\), \(\ell\ell(t)=1+\log(\ell(t))\), and \(\ell^{\mathbb A}\), \(\ell\ell^{\mathbb B}\) are the so-called broken-logarithmic functions defined by \(\ell^{\mathbb A}(t)=\ell^{\alpha_0}(t)\) for \(0<t\leq 1\), \(\ell^{\mathbb A}(t)=\ell^{\alpha_\infty}(t)\) for \(1<t<\infty\) (similarly for \(\ell\ell^{\mathbb B}\)).
The aim of this paper is to collect information about these spaces which has been obtained during several years and scattered in the literature. The authors present a useful self-contained rather comprehensive “primer” on GLZ spaces. The topics treated include embeddings among the GLZ spaces, a complete characterization of associate spaces of GLZ spaces, a full characterization of those GLZ spaces which are rearrangement-invariant Banach function spaces, and an analysis of the problem when a GLZ space \(L_{p,q;\mathbb A,\mathbb B}\) or \(L_{(p,q;\mathbb A,\mathbb B)}\) coincides with an appropriate Orlicz space, and a characterization of GLZ spaces with absolutely continuous norm.


46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
26D15 Inequalities for sums, series and integrals
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