Trudinger's inequality and continuity of potentials on Musielak-Orlicz-Morrey spaces
type: Item
instance of: scholarly article
title: Trudinger's inequality and continuity of potentials on Musielak-Orlicz-Morrey spaces
zbMATH Open document ID: 1268.46024
author: Fumi-Yuki Maeda
author: Yoshihiro Mizuta
author: Takao Ohno
author: Tetsu Shimomura
published in: Potential Analysis
publication date: 2013-02-15T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 6136718
zbMATH Keywords: Musielak-Orlicz-Morrey space
zbMATH Keywords: Trudinger inequality
zbMATH Keywords: variable exponent
reviewed by: Julian Musielak
MaRDI profile type: Publication
full work available at URL: s11118-012-9284-y
OpenAlex ID: W2069042120
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DOI: 10.1007/S11118-012-9284-Y
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description: scientific article; zbMATH DE number 6136718