Let \((X,\mu)\) be a measure space and \(\frac1p+\frac1q=1\) with \(1\leq p,q\leq\infty\). The full statements are on Inequalities.
| Quantity bounded | Inequality | Statement |
|---|---|---|
| \(\int \abs{fg}\) | Hölder | \(\norm{fg}_1 \leq \norm f_p \norm g_q\) |
| \(\norm{f+g}_p\), \(1\leq p\leq\infty\) | Minkowski | \(\norm{f+g}_p\leq\norm f_p+\norm g_p\) |
| \(\mu(\theset{\abs f > t})\), \(0<p<\infty\) | Chebyshev | \(\mu(\theset{\abs f > t}) \leq t^{-p}\norm f_p^p\) |
| \(\varphi\qty(\int f)\), \(\varphi\) convex, \(\mu(X)=1\) | Jensen | \(\varphi\qty(\int f) \leq \int \varphi\circ f\) |
| \(\norm{f * g}_r\) on \(\RR^n\) | Young | \(\norm{f*g}_r\leq\norm f_p\norm g_q\) for \(\frac1r = \frac1p+\frac1q-1\), \(1\leq p,q,r\leq\infty\) |
Consequences of Hölder’s inequality
Inclusions. If \(\mu(X)<\infty\) and \(p < q\), Hölder’s inequality applied to \(\abs f^p\cdot 1\) gives \(L^q \subseteq L^p\). For \(\mu(X)=\infty\) the inclusion can fail: on \((1,\infty)\) with Lebesgue measure, \(x^{-1}\in L^2\setminus L^1\), and on \(\RR\), \(\frac{\sin x}{x}\in L^2(\RR)\setminus L^1(\RR)\).
Interpolation. If \(0<p<r<q\leq\infty\) and \(\frac1r = \frac\theta p + \frac{1-\theta}q\) with \(\theta\in(0,1)\), then \(\norm f_r \leq \norm f_p^{\theta}\norm f_q^{1-\theta}\), by Hölder’s inequality applied to \(\abs f^{r\theta}\cdot\abs f^{r(1-\theta)}\) with exponents \(\frac{p}{r\theta}\) and \(\frac{q}{r(1-\theta)}\).
Duality. For \(g\in L^q\), \(\abs{\int fg}\leq\norm f_p\norm g_q\), so \(f\mapsto\int fg\) is a bounded linear functional on \(L^p\) of norm at most \(\norm g_q\).
For \(1<p<\infty\) and \(\norm f_p,\norm g_q\in(0,\infty)\), equality holds in Hölder’s inequality if and only if \(\abs f^p/\norm f_p^p = \abs g^q/\norm g_q^q\) almost everywhere.
Chebyshev’s inequality
A bound on \(\norm f_p\) bounds the measure of \(\theset{\abs f>t}\). Combined with the Borel–Cantelli lemma: if \(\sum_n\norm{f_n}_p^p<\infty\), then \(f_n\to0\) almost everywhere.
Particular exponents
\(p = 1\): for \(\sigma\)-finite \(\mu\), \((L^1)^* \cong L^\infty\); for Lebesgue measure on \([0,1]\) the natural map \(L^1\to(L^\infty)^*\) is not surjective.
\(p = 2\): \(L^2\) is a Hilbert space with \(\inner fg\coloneqq\int f\bar g\); if \(X\) contains disjoint sets of positive finite measure, \(L^p\) is a Hilbert space only for \(p=2\).
\(p = \infty\): \(\norm f_\infty\) is the essential supremum, the least \(M\) with \(\abs f\leq M\) almost everywhere.
\(0<p<1\): \(\norm f_p\coloneqq\qty{\int\abs f^p}^{1/p}\) is not a norm; on \([0,2]\), \(\norm{\chi_{[0,1]}+\chi_{[1,2]}}_p = 2^{1/p}>2\).