List of Inequalities

Introduction

This article lists the inequalities the corpus states and uses. An inequality is a relation of order or of size between quantities, and a row below names one inequality, records what it bounds — an inner product by a product of norms, an integral by a product of integrals, a function by its derivative, a solution by its data — and points to the article that proves it. Every row points to an article; this article introduces nothing and proves nothing. The inequalities are grouped by the layer that introduces them: the pointwise inequalities of convexity and order, the integral inequalities of the $L^p$ theory, the inequalities of the Sobolev scale, and the inequalities of geometry and of differential equations. The rows that record a failure — the converse that is false, the exponent that is sharp, the equality case that is the boundary of the inequality — stand beside the inequalities as non-examples.

The Pointwise Inequalities of Convexity and Order

Object	What it bounds, and its relation to the others	Introduced in
Cauchy–Schwarz	$\lvert\langle x,y\rangle\rvert \leq \lVert x\rVert\lVert y\rVert$; the basic inequality of an inner product space, with equality exactly on proportional vectors	Banach and Hilbert Spaces
Jensen's inequality	$\varphi(\int f\,d\mu) \leq \int \varphi\circ f\,d\mu$ for convex $\varphi$; the definition of convexity in its integral form	Convex Analysis; Measure-Theoretic Probability
the Fenchel–Young inequality	$\langle x,y\rangle \leq f(x) + f^*(y)$; the general form of Young's inequality, with equality exactly when $y \in \partial f(x)$	Convex Analysis
the arithmetic–geometric mean inequality	$\sqrt{ab} \leq (a+b)/2$; the case $p = q = 2$ of Young	Sobolev Spaces and Weak Solutions
the convexity of the sublevel sets, and the supporting hyperplane inequality	$f(x) \geq f(x_0) + \langle v, x - x_0\rangle$ for $v$ in the subdifferential	Convex Analysis
Kantorovich's inequality	the ratio $\langle Ax,x\rangle\langle A^{-1}x,x\rangle \leq (M+m)^2/(4Mm)$ for positive spectra in $[m,M]$	Nonlinear Functional Analysis

The Integral Inequalities of the $L^p$ Theory

Object	What it bounds	Introduced in
Hölder's inequality	$\int \lvert fg\rvert \leq \lVert f\rVert_p\lVert g\rVert_q$ for $1/p+1/q=1$; the fundamental inequality of the $L^p$ spaces	Measure Theory and Integration
Minkowski's inequality	$\lVert f+g\rVert_p \leq \lVert f\rVert_p + \lVert g\rVert_p$; the triangle inequality that makes $L^p$ a normed space	Measure Theory and Integration
Young's convolution inequality	$\lVert f*g\rVert_r \leq \lVert f\rVert_p\lVert g\rVert_q$ for $1/p+1/q = 1/r + 1$	Real Harmonic Analysis
Hausdorff–Young	$\lVert\hat f\rVert_q \leq \lVert f\rVert_p$ for $1 \leq p \leq 2$, $1/p + 1/q = 1$	Fourier Analysis on Euclidean Spaces
the Riesz–Thorin interpolation inequality	$\lVert T\rVert_{L^p\to L^q} \leq \lVert T\rVert_{L^{p_0}\to L^{q_0}}^{1-\theta}\lVert T\rVert_{L^{p_1}\to L^{q_1}}^{\theta}$	Interpolation Theory; Fourier Analysis on Euclidean Spaces
Chebyshev's inequality	$\mu(\lvert f\rvert \geq \lambda) \leq \lambda^{-p}\lVert f\rVert_p^p$; the bridge from $L^p$ convergence to convergence in measure	Measure Theory and Integration
the Hardy–Littlewood maximal inequality	$\lVert Mf\rVert_p \leq C_p\lVert f\rVert_p$ for $1	Real Harmonic Analysis
the Hardy–Littlewood–Sobolev inequality	the fractional integral $I_\alpha f$ is bounded from $L^p$ to $L^q$ with $1/q = 1/p - \alpha/n$	Real Harmonic Analysis
the maximal inequality for martingales	the maximal-function bound $\lambda\mathbb P(\sup_n \lvert M_n\rvert \geq \lambda) \leq \mathbb E\lvert M_N\rvert$	Martingales
the Cauchy–Schwarz inequality for integrals	$\left(\int fg\right)^2 \leq \int f^2\int g^2$; the special case $p = q = 2$ of Hölder	Fourier Analysis on Euclidean Spaces

A warning belongs here. The classical Hardy inequality, which bounds the mean of a function by the mean of its primitive, is not introduced anywhere in Parts I to III, and it is therefore absent from this list. The Hardy inequalities the corpus does carry are the maximal inequality and the fractional-integral inequality of the table above, both of them harmonic-analytic rather than the elementary averaging inequality, and they are the entries a reader who looks for "Hardy" should find.

The Inequalities of the Sobolev Scale

Object	What it bounds	Introduced in
the Sobolev embedding inequality	$\lVert f\rVert_{L^{p^}} \leq C\lVert\nabla f\rVert_{L^p}$ for $p^ = np/(n-p)$, $p < n$	Sobolev Spaces and Weak Solutions
the Sobolev inequality of geometric measure theory	the same bound as an isoperimetric inequality for the gradient measure	Geometric Measure Theory
the Gagliardo–Nirenberg inequality	$\lVert D^j f\rVert \leq C\lVert D^k f\rVert^\theta\lVert f\rVert^{1-\theta}$; the interpolation between derivatives	Sobolev Spaces and Weak Solutions
the Poincaré inequality	$\int_\Omega \lvert f - f_\Omega\rvert^p \leq C\int_\Omega \lvert\nabla f\rvert^p$; the control of a function by its gradient	Sobolev Spaces and Weak Solutions; Markov Chains and Processes
the Ladyzhenskaya inequality	$\lVert f\rVert_{L^4}^2 \leq C\lVert f\rVert_{L^2}\lVert\nabla f\rVert_{L^2}$ in two and three dimensions	Sobolev Spaces and Weak Solutions
the Morrey inequality	the Hölder continuity of a Sobolev function with $p>n$	Sobolev Spaces and Weak Solutions
the Nash inequality	$\lVert f\rVert_2^{2+4/n} \leq C\lVert\nabla f\rVert_2^2\lVert f\rVert_1^{4/n}$; the embedding that underlies the heat-kernel bounds	Random Walks on Groups
the Besov embedding	$B^s_{p,q} \hookrightarrow B^{s'}_{p',q'}$ when $s - n/p = s' - n/p'$ and $q \leq q'$	Besov and Triebel–Lizorkin Spaces
the Gagliardo–Nirenberg–Sobolev comparison	the Sobolev, Hölder, Morrey and Besov embeddings as the cases of one family	Interpolation Theory

The Inequalities of Geometry and of Differential Equations

Object	What it bounds	Introduced in
the isoperimetric inequality	$\lvert\partial E\rvert \geq n\omega_n^{1/n}\lvert E\rvert^{(n-1)/n}$; the perimeter of a set by its volume	Geometric Measure Theory
the Brunn–Minkowski inequality	$\lvert E + F\rvert^{1/n} \geq \lvert E\rvert^{1/n} + \lvert F\rvert^{1/n}$	Geometric Measure Theory
the Bellman–Grönwall inequality	a function bounded by an integral of itself is bounded by the exponential of the integral	Ordinary Differential Equations
the continuous-dependence estimate	the distance between two solutions bounded by the Grönwall factor times the perturbation	Ordinary Differential Equations
the Bellman–Grönwall inequality for delay equations	the same bound in the presence of a memory term	Delay and Functional Differential Equations
the comparison inequality for the delay equation	a solution bounded above by the solution of the associated comparison equation	Delay and Functional Differential Equations
the energy-decay inequality for the heat semigroup	$\lVert T(t)x\rVert \leq e^{\omega t}\lVert x\rVert$, the semigroup growth bound	Semigroups and Evolution Equations
the Ekeland variational inequality	the perturbed minimisation inequality $f(v) \leq \inf f + \epsilon$ at a near-minimiser	Nonlinear Functional Analysis

Failures, Sharpness and Equality Cases

Object	The failure or the sharpness it records	Introduced in
Hölder's inequality outside the conjugate exponents	fails if $1/p + 1/q \neq 1$; equality holds exactly when $\lvert f\rvert^p$ and $\lvert g\rvert^q$ are proportional	Measure Theory and Integration
Young's convolution inequality at the endpoint $p = q = 1$	is an equality, $\lVert f*g\rVert_1 = \lVert f\rVert_1\lVert g\rVert_1$, the boundary of the interpolated family	Real Harmonic Analysis
the sharp constant of the Sobolev inequality	attained exactly by the translates and dilates of a fixed extremal, so the inequality is strict off that orbit	Sobolev Spaces and Weak Solutions
the maximal inequality at $p = 1$	the strong-type bound fails; only the weak-type $(1,1)$ bound holds	Real Harmonic Analysis
the isoperimetric inequality	equality holds exactly for balls; no other set attains it	Geometric Measure Theory
the Poincaré inequality without a normalisation	fails on functions with a constant part; the mean must be removed for the inequality to hold	Sobolev Spaces and Weak Solutions
Jensen's inequality for a non-convex $\varphi$	fails; convexity is sharp for the inequality	Convex Analysis

Summary

This list gathers the inequalities of the corpus: the pointwise inequalities of convexity — Cauchy–Schwarz, Jensen, Young and the arithmetic–geometric mean — the integral inequalities of the $L^p$ theory, among them Hölder, Minkowski, Hausdorff–Young and the maximal and fractional-integral bounds, the inequalities of the Sobolev scale with the Poincaré, Gagliardo–Nirenberg, Ladyzhenskaya and Nash bounds, and the isoperimetric, Brunn–Minkowski and Bellman–Grönwall inequalities of geometry and of differential equations. The closing table records the sharpness of each and the cases where it fails.

Summary of Notation

The objects are named rather than denoted; the symbols in the tables are collected here.

Symbol	Meaning
$p$, $q$	conjugate exponents, $1/p + 1/q = 1$
$\lVert f\rVert_p$	the $L^p$ norm
$\nabla f$, $D^j f$	gradient and $j$-th derivative
$Mf$	the Hardy–Littlewood maximal function
$I_\alpha f$	the Riesz fractional integral of order $\alpha$
$p^*$	the Sobolev conjugate $np/(n-p)$
$E$, $F$	measurable sets; $\lvert E\rvert$ their volume