Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 23F / b

Solution

For

1 \leq p < \infty

, choose nonzero

ϕ \in C_{c} (R^{n - 1})

and

ψ \in C_{c} (R)

with

ψ (0) = 1

u_{k} (x^{'}, x_{n}) = ϕ (x^{'}) ψ (k x_{n}) .

These functions are continuous and belong to the Lebesgue space

L^{p} (R^{n})

, while their restrictions to the hyperplane are all the same:

u_{k} (x^{'}, 0) = ϕ (x^{'}) .

Changing variables in the normal coordinate gives

∥ u_{k} ∥_{L^{p} (R^{n})}^{p} = \frac{1}{k} ∥ ϕ ∥_{L^{p} (R^{n - 1})}^{p} ∥ ψ ∥_{L^{p} (R)}^{p} ⟶ 0.

If the claimed bounded

T

existed, then

0 < ∥ ϕ ∥_{p} = ∥ T u_{k} ∥_{p} \leq ∥ T ∥ ∥ u_{k} ∥_{p} ⟶ 0,

a contradiction. This scaling argument is the failure of an Lp hyperplane trace.

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