 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-107/3/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

L^{p, μ} (Ω)

consists of

f \in L^{p} (Ω)

for which

[f]_{L^{p, μ}}^{p} = sup_{x_{0}, r} r^{- μ} \int_{Ω \cap B_{r} (x_{0})} ∣ f - f_{Ω \cap B_{r} (x_{0})} ∣^{p} < \infty,

where

f_{E} = ∣ E ∣^{- 1} \int_{E} f

. On a smooth bounded domain,

L^{p, n + p α} (Ω) = C^{0, α} (\overline{Ω})

with equivalent norms for

0 < α \leq 1

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