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Equicontinuity forces convergence in H^s-space.

Source: Alibaba Global Math Competition 2021, Problem 7

July 4, 2021
Convergencesobolev spacecontinuitycollege contests

Problem Statement

A subset Q⊂Hs(R)Q \subset H^s(\mathbb{R}) is said to be equicontinuous if for any ε>0\varepsilon>0, ∃δ>0\exists \delta>0 such that \|f(x+h)-f(x)\|_{H^s}<\varepsilon,   \forall \vert h\vert<\delta,   f \in Q. Fix r<sr<s, given a bounded sequence of functions fn∈Hs(Rf_n \in H^s(\mathbb{R}. If fnf_n converges in Hr(R)H^r(\mathbb{R}) and equicontinuous in Hs(R)H^s(\mathbb{R}), show that it also converges in Hs(R)H^s(\mathbb{R}).
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