Hölder's Inequality
Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $p, q \in \R_{>0}$ such that $\dfrac 1 p + \dfrac 1 q = 1$.
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Let $f \in \map {\LL^p} \mu, f: X \to \R$, and $g \in \map {\LL^q} \mu, g: X \to \R$, where $\LL$ denotes Lebesgue space.
Then their pointwise product $f g$ is $\mu$-integrable, that is:
- $f g \in \map {\LL^1} \mu$
and:
- $\ds \size {f g}_1 = \int \size {f g} \rd \mu \le \norm f_p \cdot \norm g_q$
where the $\norm {\, \cdot \,}_p$ signify $p$-seminorms.
Equality
Equality, that is:
- $\ds \int \size {f g} \rd \mu = \norm f_p \cdot \norm g_q$
holds if and only if, for almost all $x \in X$:
- $\dfrac {\size {\map f x}^p} {\norm f_p^p} = \dfrac {\size {\map g x}^q} {\norm g_q^q}$
Hölder's Inequality for Sums
Let $p, q \in \R_{>0}$ be strictly positive real numbers such that:
- $\dfrac 1 p + \dfrac 1 q = 1$
Let:
- $\mathbf x = \sequence {x_n} \in \ell^p$
- $\mathbf y = \sequence {y_n} \in \ell^q$
where $\ell^p$ denotes the $p$-sequence space.
Let $\norm {\mathbf x}_p$ denote the $p$-norm of $\mathbf x$.
Then $\mathbf x \mathbf y = \sequence {x_n y_n} \in \ell^1$, and:
- $\norm {\mathbf x \mathbf y}_1 \le \norm {\mathbf x}_p \norm {\mathbf y}_q$
Generalized Hölder Inequality
Let $\struct {X, \Sigma, \mu}$ be a measure space.
For $i = 1, \ldots, n$ let $p_i \in \R_{>0}$ such that:
- $\ds \sum_{i \mathop = 1}^n \frac 1 {p_i} = 1$
Let $f_i \in \map {\LL^{p_i} } \mu, f_i: X \to \R$, where $\LL$ denotes Lebesgue space.
Then their pointwise product $\ds \prod_{i \mathop = 1}^n f_i$ is integrable, that is:
- $\ds \prod_{i \mathop = 1}^n f_i \in \map {\LL^1} \mu$
and:
- $\ds \norm {\prod_{i \mathop = 1}^n f_i}_1 = \int \size {\prod_{i \mathop = 1}^n f_i} \rd \mu \le \prod_{i \mathop = 1}^n \norm {f_i}_{p_i}$
where the various instances of $\norm {\, \cdot \,}$ signify $p$-seminorms.
Proof
Let $x \in X$.
Let:
- $a_x := \dfrac {\size {\map f x} } {\norm f_p}$
and:
- $b_x := \dfrac {\size {\map g x} } {\norm g_q}$
Applying Young's Inequality for Products to $a_x$ and $b_x$:
- $\dfrac {\size {\map f x \map g x} } {\norm f_p \cdot \norm g_q} \le \dfrac {\size {\map f x}^p} {p \norm f_p^p} + \dfrac {\size {\map g x}^q} { q \norm g_q^q}$
By Integral of Positive Measurable Function is Monotone, integrating both sides of this inequality over x yields:
- $\ds \dfrac {\int \size {\map f x \map g x} \map \mu {\d x} } {\norm f_p \cdot \norm g_q} \le \frac {\norm f_p^p} {p \norm f_p^p} + \frac {\norm g_p^q} {q \norm g_q^q} = \frac 1 p + \frac 1 q = 1$
so:
- $\ds \int \size {\map f x \map g x} \map \mu {\d x} \le \norm f_p \cdot \norm g_q$
If we have equality, then:
- $\ds \int \paren {\frac {\size {\map f x}^p} {p \norm f_p^p} + \frac {\size {\map g x}^q} {q \norm g_q^q} - \frac {\size {\map f x \map g x} } {\norm f_p \cdot \norm g_q} } \map \mu {\d x} = 0$
As:
- $\dfrac {\size {\map f x}^p} {p \norm f_p^p} + \dfrac {\size {\map g x}^q} {q \norm g_q^q} - \dfrac {\size {\map f x \map g x} } {\norm f_p \cdot \norm g_q} \ge 0$
it follows from Measurable Function Zero A.E. iff Absolute Value has Zero Integral that:
- $\dfrac {\size {\map g x}^p} {p \norm f_p^p} + \dfrac {\size {\map g x}^q} {q \norm g_q^q} = \dfrac {\size {\map f x \map g x} } {\norm f_p \cdot \norm g_q}$ a.e.
By Young's Inequality for Products, we have equality if and only if $b_x = {a_x}^{p - 1}$.
Raising both sides to the $q$th power gives:
- $\dfrac {\size {\map g x}^p} {\norm f_p^p} = \dfrac {\size {\map g x}^q} {\norm g_q^q}$
as $\paren {p - 1} q = p$.
$\blacksquare$
Source of Name
This entry was named for Otto Ludwig Hölder.
Sources
- 1991: David Williams: Probability with Martingales ... (previous) ... (next): $6.12$: The 'elementary formula' for expectation
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $12.2$