ϕ (x) = \int_{0}^{x} u (t) d t, A u (x) = \frac{ϕ ( x )}{x} .

∣ ϕ (x) ∣^{p} x^{1 - p} \leq \int_{0}^{x} ∣ u (t) ∣^{p} d t ⟶ 0 (x ↓ 0) .

Using

x^{- p} d x = - (p - 1)^{- 1} d (x^{1 - p})

and integration by parts, while discarding the nonpositive boundary term at

x = 1

, yields

∥ A u ∥_{p}^{p} = \int_{0}^{1} ∣ ϕ ∣^{p} x^{- p} d x \leq \frac{p}{p - 1} \int_{0}^{1} ∣ ϕ ∣^{p - 1} x^{1 - p} ∣ u ∣ d x = \frac{p}{p - 1} \int_{0}^{1} ∣ A u ∣^{p - 1} ∣ u ∣ d x .

Another application of Hölder's inequality gives

∥ A u ∥_{p}^{p} \leq \frac{p}{p - 1} ∥ A u ∥_{p}^{p - 1} ∥ u ∥_{p} .

After cancellation, with the zero case immediate,

∥ A u ∥_{L^{p} (0, 1)} \leq \frac{p}{p - 1} ∥ u ∥_{L^{p} (0, 1)} .

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