Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 23G / a / Solution

Tonelli's theorem and the triangle inequality give

\int_{0}^{\infty} ∣ G (y) g (y) ∣ d y \leq \int_{0}^{\infty} \int_{0}^{\infty} ∣ F (x, y) g (y) ∣ d x d y = \int_{0}^{\infty} (\int_{0}^{\infty} ∣ F (x, y) g (y) ∣ d y) d x .

Applying Holder inequality in the

y

variable for each fixed

x

yields

\int_{0}^{\infty} ∣ G (y) g (y) ∣ d y \leq \int_{0}^{\infty} (\int_{0}^{\infty} ∣ F (x, y) ∣^{p} d y)^{1/ p} ∥ g ∥_{q} d x,

and

∥ g ∥_{q} \leq 1

gives the claimed inequality. This is the duality proof of the Minkowski integral inequality.

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