Some properties of Orlicz-Lorentz spaces (Q651829)

Let \(\varphi\) be an Orlicz function and let \(\omega\) be a weight function. The Orlicz-Lorentz space \(\Lambda^{\mathbb{R}}_{\varphi, \omega}\) is defined as the space of locally integrable functions \(f\) in \(\mathbb{R}\) such that \[ \int^\infty_0 \varphi(\lambda f^*(x))\omega(x)\, dx< \infty \] for some \(\lambda> 0\), where \(f^*\) is the nonincreasing rearrangement of \(f\). The paper is devoted to the investigation of relations between the Orlicz norm \(\| f\|^1_{\Lambda^{\mathbb{R}}_{\varphi, \omega}}\) and the Luxemburg norm \(\| f\|_{\Lambda^{\mathbb{R}}_{\varphi,\omega}}\), which are equivalent. In particular, the best constants \(C_1,C_2>0\) for which \[ C_1\| f\|_{\Lambda^{\mathbb{R}}_{\varphi,\omega}}\leq\| f\|^1_{\Lambda^{\mathbb{R}}_{\varphi,\omega}}\leq C_2\| f\|_{\Lambda^{\mathbb{R}}_{\varphi,\omega}} \] for all \(f\in\Lambda^{\mathbb{R}}_{\varphi,\omega}\) are established. This is applied to obtain a version of the Landau-Kolmogorov inequality in the spaces \(\Lambda_{\varphi,\omega}\).