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

Abstract.

Let $a_j : {\mathbb{R}} \rightarrow {\mathbb{R}}$ be a sequence of Borel measurable functions satisfying, for a function $K \in L_{\rm loc}^{1}, K : {\mathbb{R}} \rightarrow [1,\infty),$ the inequalities

$$ 1/K(x) \leq a_j (x) \leq K(x)\quad {\rm a.e.}\, x \in {\mathbb{R}} $$

and suppose

$$ a_j \rightharpoonup a\quad {\rm in}\,\sigma(L^1, L^\infty). $$

Then there exists a sequence of increasing homeomorphisms $h_j : {\mathbb{R}} \rightarrow {\mathbb{R}}$ converging to a homeomorphism $h : {\mathbb{R}} \rightarrow {\mathbb{R}}$ weakly in $W^{1,1}_{\rm loc}({\mathbb{R}})$ and locally uniformly, such that

$$ 1/a_j(h^{-1}_{j}) \rightharpoonup 1/a(h^{-1})\quad{\rm in}\,\sigma(L^{1}, L^{\infty}). $$