Trudinger's inequality and continuity of potentials on Musielak-Orlicz-Morrey spaces
Let \(G\) be a bounded, open set in \(\mathbb{R}^N\) with diameter \(d_G\). The Musielak-Orlicz-Morrey space \(L^{\phi,\kappa}(G)\) is defined as the space of locally integrable functions \(f\) on \(G\) for which the norm \[ \| f\|_{L^{\phi,\kappa}(G)}= \text{inf}\Biggl\{\lambda> 0: \sup_{x\in G,\,0< r< d_G}|B(x, r)|^{-1} \kappa(x,r) \int_{G\cap B(x,r)} \overline\phi(y,|f(y)|\lambda^{-1})\,dy\leq 1\Biggr\} \] is finite. Here, the function \(\overline\phi: G\times [0,\infty)\to [0,\infty)\) obeys a number of auxiliary conditions. The authors consider a potential kernel \(J(x,r): G\times (0,d_G)\to (0,\infty)\) satisfying some conditions. The main result consists in proving a respective version of Trudinger's inequality for the norm \[ \|\cdot\|_{L^{\phi,\kappa}(G)}: {\lambda_\varepsilon(z,r)\over |B(z,r)|} \int_{G\cap B(z,r)} \psi\Biggl(x,Jf(x){1\over C^*}\Biggr)\,dx\leq 1, \] where \(\psi\) satisfies suitable conditions.
scientific article; zbMATH DE number 6136718
scientific article