 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-8/2/c/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

s

t

(U_{a} (t, s) g) (x, v) = e^{- \int_{s}^{t} a (r, x - v (t - r), v) d r} g (x - v (t - s), v) .

By the argument in (b),

∥ U_{a} (t, s) g ∥_{2} \leq ∥ g ∥_{2}

τ f (t) = \int_{0}^{t} U_{a} (t, s) K_{s} f (s) d s .

∥ τ f (t) ∥_{2} \leq C \int_{0}^{t} ∥ f (s) ∥_{2} d s .

Therefore the requested estimate is

∥ τ f (t) ∥_{2} \leq Ct 0 \leq s \leq t sup ∥ f (s) ∥_{2} .

For strongly measurable bounded

L^{2}

-valued

f

, these are Bochner integrals; their finite norm bounds establish existence of the integrals.

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