Grand quasi Lebesgue spaces. (English) Zbl 1481.46022
Let \(\psi_{\alpha,\beta}\) be a continuous strictly positive function on the interval \((\alpha, \beta)\) with \(0<\alpha<\beta\le 1\) such that \(\inf_{p\in(\alpha,\beta)}\psi_{\alpha,\beta}(p)>0\). The authors introduce the grand Lebesgue space \(G\psi_{\alpha,\beta}\) as the collection of all measurable functions \(f\) defined on a measure space such that \(\|f\|=\sup_{p\in(\alpha,\beta)}\frac{\|f\|_p}{\psi_{\alpha,\beta}(p)}<\infty\), where \(\|f\|_p\) stands for the \(L^p\)-quasi-norm of \(f\). Several basic results on the space \(G\psi_{\alpha,\beta}\) are established. It is shown that \(G\psi_{\alpha,\beta}\) is a non-locally convex \(F\)-space, it is rearrangement-invariant, its associate and dual spaces are trivial, its Boyd indices \(\gamma_1,\gamma_2\) satisfy \(1/\beta<\gamma_1\le\gamma_2<1/\alpha\). An extension of the Hardy inequality for \(G\psi_{\alpha,\beta}\) is also proved.
Reviewer: Oleksiy Karlovych (Lisboa)
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
Keywords:
Lebesgue-Riesz spaces; quasi-Banach spaces; grand quasi Lebesgue spaces; tail function; contraction principle; Hardy operatorsReferences:
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