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

Authors:

Gioconda Moscariello Carlo Sbordone

Institution:

(1) Dipartimento di Matematica “R. Caccioppoli”, Universitá di Napoli “Federico II”, Via Cintia, 80126 Napoli, Italy

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

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

Dedicated to the memory of Jean Leray

Keywords:

Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 26A46 28A20 30C62

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