A direct approach to the duality of grand and small Lebesgue spaces
Let \(\mathcal{M}_{0}\) be the set of all Lebesgue measurable function in the interval \((0,1)\), finite a.e.\ in it and let \(\mathcal{M}_{0}^{+}\) be the class of all nonnegative functions of \(\mathcal{M}_{0}\). For \(f \in \mathcal{M}_{0}\), the decreasing rearrangement \(f^{*}\) of \(f\) is defined by \[ f^{*} = \inf\{ \lambda > 0: |\{x \in (0,1): |f(x)| > \lambda\}| \leq t \}, \] where the symbol \(|E|\) denotes the Lebesgue measure of \(E\). Let \(1 < p < \infty\) and \(\Theta > 0\). For all \(f \in \mathcal{M}_{0}\), we set \[ |\lceil f \rceil|_{p,\Theta} := \sup_{0<t<p}(1 - \log t) ^{-\frac{\Theta}{p}} \left(\int_{0}^{1}f^{*}(s)^{p}\,ds \right)^{\frac{1}{p}} \] and \[ \|f\|_{p,\Theta} := \sup_{0<\varepsilon<p-1}\varepsilon^{\Theta} \left(\int_{0}^{1} |f^{*}(x)|^{p-\varepsilon}\,dx \right)^{\frac{1}{p-\varepsilon}}. \] The authors prove that \[ c_{1}\|f\|_{p,\Theta} \leq |\lceil f \rceil|_{p,\Theta} \leq \|f\|_{p,\Theta} \] for some \(c_{1}, c_{2} > 0\) and for every \(f \in \mathcal{M}_{0}^{+}\). A Hölder type inequality is also proved.
scientific article; zbMATH DE number 5532904
scientific article