Some inequalities in classical spaces with mixed norms
Let \((T_1,\Sigma_1,\mu_1)\) and \((T_2,\Sigma_2,\mu_2)\) be nonatomic measure spaces with \(\mu_1(T_1)= \mu_2(T_2)= 1\) and \((T,\Sigma,\mu)\) be their product. Let \(E\) and \(\widetilde E\) be Banach function spaces on \((T_2,\Sigma_2,\mu_2)\) and \((T,\Sigma,\mu)\), respectively. We say that the \(L^\infty\)-mixed norm property (\(L^\infty\)-MNP) holds if there exists a constant \(C> 0\) such that \[ \| K(s, t)\|_{\widetilde E(s,t)}\leq C\,\underset{t\in T_1}{}{\text{ess\,sup}}\| K(s,t)\|_{E,s} \] for every \(K\in \widetilde E\). The main result of the present paper is that in the Lorentz space \(\Lambda(t^\alpha, p)\) \((0< \alpha< 1,1\leq p<\infty)\) the \(L^\infty\)-MNP does not hold.
mixed norm inequalities for functions of many variables
scientific article; zbMATH DE number 1880380