A note on noneffective weights in variable Lebesgue spaces (Q657336)

Summary: We study noneffective weights in the framework of variable exponent Lebesgue spaces, and we show that \(L^{p(\cdot)} (\Omega) = L^{p(\cdot)}_\omega (\Omega)\) if and only if \(\omega(x)^{1/p(x)} \sim \text{constant}\) in the set where \(p(\cdot) < \infty\), and \(\omega(x) \sim \text{constant}\) in the set where \(p(\cdot) = 0\).