Hölder's inequalities and multilinear singular integrals on generalized Orlicz spaces (Q6618752)

In the paper, a generalized version of Hölder's inequality in the framework of generalized Orlicz spaces is stated. The inequality in question is \N\[\N\big\|\prod_{j=1}^m f_j \big\|_{L^{\varphi(\cdot)}(\mathbb{R}^n)}\le C \prod_{j=1}^m \|f_j \|_{L^{\varphi_j(\cdot)}(\mathbb{R}^n)}. \N\]\NHere \(\varphi_j\in \Phi_w(\mathbb{R}^n)\) for \(j=1,\ldots, m\), \(\Phi_w(\mathbb{R}^n)\) stand for the so-called class of generalized (weak) \(\Phi\)-functions, and \(\varphi=(\prod_{j=1}^m \varphi_j^{-1})^{-1}\) (appropriately defined) is the so-called left-inverse. Moreover, \(\|\cdot\|_{L^{\varphi(\cdot)}(\mathbb{R}^n)}\) and \(\|\cdot\|_{L^{\varphi_j(\cdot)}(\mathbb{R}^n)}\) denote the norms in the generalized Orlicz spaces accordingly defined for functions \(\varphi\) and \(\varphi_j\), respectively.