User:Addem/Holder

Hölder's Inequality

There are two theorems which go by the title of Hölder's Inequality, one dealing with summation and the other with integration with respect to a measure.

In both cases it is asserted that, when $1 \le p \le \infty$ and $q$ is the exponential conjugate of $p$, then

$\norm {a b}_1 \le \norm a_p \norm b_q$

In the summation case we interpret $\norm {\, \cdot \,}_p$ as the $p$-norm and in the integral case we interpret it as the $p$-seminorm.

The result for summation can be deduced from the result for integration, using the counting measure. It also has independent proofs.

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:

$\norm {\mathbf x \mathbf y}_1 \le \norm {\mathbf x}_p \norm {\mathbf y}_q$

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $p, q \in \R$ such that $\dfrac 1 p + \dfrac 1 q = 1$.

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.

$f g \in \map {\LL^1} \mu$

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.

$\ds \int \size {f g} \rd \mu = \norm f_p \cdot \norm g_q$

$\dfrac {\size {\map f x}^p} {\norm f_p^p} = \dfrac {\size {\map g x}^q} {\norm g_q^q}$

This entry was named for Otto Ludwig Hölder.