Convergence in variable Lebesgue spaces
Given a measurable set \(\Omega\subset\mathbb R^n\), an exponent function is a measurable function \(p(\cdot): \Omega\to [1,\infty] \). Let \(p_-= \text{ess\,inf}_{x \in \Omega}\, p(x)\) and \(p_+= \text{ess\,sup}_{x \in\Omega}\, p(x)\). We define the modular functional \[ \rho(f)= \int_\Omega |f(x)|^{p(x)}\,dx, \] and define the norm \[ \|f\|_{p(\cdot)}= \inf \{ \lambda >0: \rho(f/\lambda) leq 1 \}. \] Given a sequence \(\{ f_k \} \subset L^{p(\cdot)}(\Omega)\), we say that it converges in norm to \(f\in L^{p(\cdot)}(\Omega)\) if \(\|f - f_k\|_{p(\cdot)} \to 0\) as \(k \to \infty\). We say that \(f_k \to f\) in modular if there exists \(\beta >0\) such that \(\rho( \beta (f - f_k))\to 0\) as \(k \to \infty\). When \(p_{+} < \infty\), \textit{X. Fan} and \textit{D. Zhao} [J. Math. Anal. Appl. 263, No.~2, 424--446 (2001; Zbl 1028.46041)] proved that the following are equivalent: (1) \(f_k \to f\) in norm, (2) \(f_k \to f\) in modular, (3) \(f_k \to f \) in measure and for some \(\gamma >0\), \(\rho(\gamma f_k) \to \rho(\gamma f)\). When \(p_{+} = \infty\) the situation is much more complicated, and these three conditions are not equivalent in general. The authors examine each of these equivalences including the case \(p_{+}= \infty\). For this purpose they define \[ \rho (f) = \int_{\Omega \setminus \Omega_{\infty}}|f(x)|^{p(x)}\,dx+\|f \|_{L^{\infty}(\Omega_{\infty})}, \] where \(\Omega_{\infty} = \{ x \in \Omega : p(x) = \infty \}\).
scientific article; zbMATH DE number 5770008
scientific article