MathDB

7

Part of 2021 Alibaba Global Math Competition

Problems(1)

Equicontinuity forces convergence in H^s-space.

Source: Alibaba Global Math Competition 2021, Problem 7

7/4/2021
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}).
Convergencesobolev spacecontinuitycollege contests