 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-4/23i/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Let

S^{'} (R^{n})

s \in R

H^{s} (R^{n})

H^{s} (R^{n}) = {u \in S^{'} (R^{n}) : (1 + ∣ ξ ∣^{2})^{s /2} u (ξ) \in L^{2} (R^{n})},

with norm

∥ u ∥_{H^{s}}^{2} = \int_{R^{n}} (1 + ∣ ξ ∣^{2})^{s} ∣ u (ξ) ∣^{2} d ξ .

Changing the normalization of the Fourier transform changes this norm only by a fixed constant.

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