A note on weak convergence in \(L^1_{\text{loc}}(\mathbb R)\) (Q2472535)

Let \(a_j: \mathbb{R} \to \mathbb{R}\) be a sequence of Borel-measurable functions, \(K: \mathbb{R} \to [1,\infty), K \in L^1_{\text{loc}},\) and suppose that \[ 1/K(x) \leq a_j(x) \leq K(x) \] for a.e. \(x\in \mathbb{R}.\) The authors prove that if \(a_j \rightharpoonup a\) in \(\sigma(L^1,L^{\infty}),\) then there exists a sequence of increasing homeomorphisms \(h_j:\mathbb{R} \to \mathbb{R}\) converging to a homeomorphism \(h:\mathbb{R} \to \mathbb{R}\) weakly in \(W^{1,1}_{\text{loc}}(R)\) and locally uniformly such that \[ 1/a_j(h_j^{-1}) \rightharpoonup 1/a(h^{-1})\quad\text{in } \sigma(L ^1, L^{\infty}). \]