共轭指数与 Young 不等式
(1, ∞) 中的两个指数 p、q 当 1/p + 1/q = 1 时称为 共轭。于是 q = p/(p−1)；自共轭情形是 p = q = 2。Lᵖ 中乘积的全部理论都由这一配对支配，其种子是两个非负数之间的逐点不等式。
Young's inequality. For a, b ≥ 0 and conjugate p, q:
a·b ≤ aᵖ/p + b^q/q.
Proof (via concavity of log). If a = 0 or b = 0 it is clear.
For a, b > 0, the logarithm is concave, so for weights 1/p + 1/q = 1:
log( (1/p)·aᵖ + (1/q)·b^q )
≥ (1/p)·log(aᵖ) + (1/q)·log(b^q)
= log(a) + log(b)
= log(a·b).
Applying the increasing function exp to both sides:
(1/p)·aᵖ + (1/q)·b^q ≥ a·b. ∎Hölder 不等式
Hölder 不等式 说：若 f ∈ Lᵖ、g ∈ Lq 且 p、q 共轭，则 fg ∈ L¹ 且 ‖fg‖₁ ≤ ‖f‖ₚ · ‖g‖q。当 p = q = 2 时这正是 Cauchy–Schwarz 不等式。证明是在把两函数归一化为单位范数后对 Young 不等式积分。
Hölder. ∫ |fg| dμ ≤ ‖f‖ₚ · ‖g‖q.
If ‖f‖ₚ = 0 or ‖g‖q = 0 then fg = 0 a.e. and both sides are 0.
Otherwise set F = |f|/‖f‖ₚ , G = |g|/‖g‖q , so ‖F‖ₚ = ‖G‖q = 1.
Apply Young pointwise with a = F(x), b = G(x):
F·G ≤ Fᵖ/p + G^q/q.
Integrate over X:
∫ F·G dμ ≤ (1/p)∫Fᵖ dμ + (1/q)∫G^q dμ
= (1/p)·1 + (1/q)·1 = 1.
Multiply back through by ‖f‖ₚ·‖g‖q:
∫ |f g| dμ ≤ ‖f‖ₚ · ‖g‖q. ∎Minkowski 不等式即三角不等式
Minkowski 不等式 断言 ‖f + g‖ₚ ≤ ‖f‖ₚ + ‖g‖ₚ。这恰是 ‖·‖ₚ 的 三角不等式，是称 Lᵖ 为 赋范向量空间 所需的最后一条公理。技巧是把 |f+g|ᵖ 拆成 |f+g|·|f+g|^(p−1) 并对每块用 Hölder。
Minkowski. ‖f+g‖ₚ ≤ ‖f‖ₚ + ‖g‖ₚ (1 < p < ∞). Write |f+g|ᵖ = |f+g| · |f+g|^(p−1) ≤ (|f|+|g|)·|f+g|^(p−1). Integrate and split: ‖f+g‖ₚᵖ ≤ ∫ |f|·|f+g|^(p−1) + ∫ |g|·|f+g|^(p−1). Let q = p/(p−1) be the conjugate. Note |f+g|^(p−1) ∈ Lq because ∫ (|f+g|^(p−1))^q = ∫ |f+g|ᵖ = ‖f+g‖ₚᵖ < ∞. Apply Hölder to each term: ∫ |f|·|f+g|^(p−1) ≤ ‖f‖ₚ · ‖ |f+g|^(p−1) ‖q = ‖f‖ₚ · ‖f+g‖ₚ^(p−1). Same for g. Therefore ‖f+g‖ₚᵖ ≤ (‖f‖ₚ + ‖g‖ₚ) · ‖f+g‖ₚ^(p−1). If ‖f+g‖ₚ > 0, divide by ‖f+g‖ₚ^(p−1): ‖f+g‖ₚ ≤ ‖f‖ₚ + ‖g‖ₚ. ∎