Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 359 / 1 / a / ii

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- Solution ii

Solution

Interpolation between

L^{2}

and

L^{6}

, followed by the three-dimensional Sobolev inequality, gives

∥ v ∥_{L^{3}} \leq ∥ v ∥_{L^{2}}^{1/2} ∥ v ∥_{L^{6}}^{1/2} \leq c ∣ v ∣^{1/2} ∥ v ∥^{1/2} .

Apply this with

v = P_{N} w

and use part i:

∥ P_{N} w ∥_{L^{3}} \leq c ∣ P_{N} w ∣^{1/2} (λ_{N}^{1/2} ∣ P_{N} w ∣)^{1/2} = c λ_{N}^{1/4} ∣ P_{N} w ∣.

Orthogonal projection is contractive in

H

, so

∥ P_{N} w ∥_{L^{3}} \leq c λ_{N}^{1/4} ∣ w ∣ .

The Sobolev constant is dimensionless after using the periodic-domain normalization, so the estimate is scale invariant.

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