 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-4/23f/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Let

\overline{h}_{R} = \frac{\int _{- R}^{R} h ( y ) e ^{- Φ (y)} d y}{\int _{- R}^{R} e ^{- Φ (y)} d y}, q = h - \overline{h}_{R} .

Then

q^{'} = h^{'}

and

q

has zero weighted mean. Part (b) gives

∥ q ∥_{\infty}^{2} \leq C_{R}^{2} \int_{- R}^{R} ∣ h^{'} ∣^{2} e^{- Φ} .

Therefore, with

Z_{R} = \int_{- R}^{R} e^{- Φ}

\int_{- R}^{R} ∣ h - \overline{h}_{R} ∣^{2} e^{- Φ} \leq Z_{R} C_{R}^{2} \int_{- R}^{R} ∣ h^{'} ∣^{2} e^{- Φ} .

Thus the compact-interval Weighted Poincare inequality holds with

λ_{R} = (Z_{R} C_{R}^{2})^{- 1} > 0 .

 New to topics? Read the docs here!