Hölder's Inequality

• categories: Real Analysis, Functional analysis, Theorem

Hölder’s inequality is a fundamental result in analysis, providing an upper bound for the integral or sum of the product of two functions in terms of their $L^p$ and $L^q$ norms.

Statement

Let $(X,\Sigma ,\mu )$ be a measure space, and let $f,g:X\to \mathbb{R}$ (or $\mathbb{C}$) be measurable functions. For exponents $p,q\in [1,\infty ]$ such that $\frac{1}{p}+\frac{1}{q}=1$, we have:

$${\int }_X|f(x)g(x)|d\mu \le {\left({\int }_X|f(x){|}^{p}d\mu \right)}^{1/p}{\left({\int }_X|g(x){|}^{q}d\mu \right)}^{1/q}.$$

If $p=1$ and $q=\infty$ (or vice versa), the inequality is interpreted as:

$${\int }_X|f(x)g(x)|d\mu \le \Vert f{\Vert }_1\Vert g{\Vert }_{\infty }.$$

Special Case: Finite Sums

For $f,g\in {\mathbb{R}}^n$ with components $f=(f_1,f_2,\dots ,f_n)$ and $g=(g_1,g_2,\dots ,g_n)$, Hölder’s inequality reduces to:

$$\sum _{i=1}^n|f_i{g}_i|\le {\left(\sum _{i=1}^n|f_i{|}^p\right)}^{1/p}{\left(\sum _{i=1}^n|g_i{|}^q\right)}^{1/q}.$$

Key Cases

1. Cauchy–Schwarz Inequality ($p=q=2$): $${\int }_X|f(x)g(x)|d\mu \le {\left({\int }_X|f(x){|}^{2}d\mu \right)}^{1/2}{\left({\int }_X|g(x){|}^{2}d\mu \right)}^{1/2}.$$
2. Young’s Inequality ($p=1$, $q=\infty$): $${\int }_X|f(x)g(x)|d\mu \le \Vert f{\Vert }_1\Vert g{\Vert }_{\infty }.$$

Intuition

Hölder’s inequality bounds the integral (or sum) of the product $|fg|$ by leveraging the “spread” of $f$ and $g$ over different $L^p$ and $L^q$ norms. This inequality is essential for establishing the completeness of $L^p$ spaces and has applications across analysis, probability, and partial differential equations.

Proof Outline

1. Convexity: Use Young’s inequality for products of real numbers, which asserts that for any $a,b\ge 0$: $$ab\le \frac{a^p}{p}+\frac{b^q}{q}.$$
2. Apply to Integrals: Set $a=|f(x)|$ and $b=|g(x)|$, integrate both sides, and simplify using properties of integrals and norms