---
tag: 6GYQNB2862
kind: statement
role: theorem
name: Littlewood–Paley inequality
msc: [42B25]
uses: [H4E1PW67Z1, D1PAT0K446, SASP14MA1Z, 3SCXCW4AH3, JQ9WXC729D]
about: [H4E1PW67Z1]
sources:
  - url: https://en.wikipedia.org/wiki/Littlewood%E2%80%93Paley_theory
---

## Statement

Let $n \ge 1$ and $1 < p < \infty$, and let $Sf$ be the [[Littlewood–Paley square function|H4E1PW67Z1]] on $\R^n$, built from a fixed cut-off $\chi$.
Then there are constants $0 < c \le C$, depending only on $n$, $p$ and $\chi$, such that
$$c \norm{f}_{L^p(\R^n)} \le \norm{Sf}_{L^p(\R^n)} \le C \norm{f}_{L^p(\R^n)}$$
for every $f$ in the [[Lp space|SASP14MA1Z]] $L^p(\R^n)$.

## Remarks

For $p = 2$ the inequality follows from Plancherel's theorem.
For the upper bound, view $f \mapsto (\Delta_j f)_{j \in \Z}$ as a single operator from $L^p(\R^n)$ to $L^p(\R^n; \ell^2)$.
Its kernel $(\check\psi_j)_j$ satisfies the size and smoothness estimates of a vector-valued [[Calderón–Zygmund operator|3SCXCW4AH3]], so the $L^2$ bound extends to $1 < p < \infty$.
The lower bound follows by duality.
Let $\tilde\Delta_j = \Delta_{j-1} + \Delta_j + \Delta_{j+1}$; its multiplier equals $1$ on the support of the multiplier of $\Delta_j$, so $\int f \bar g = \sum_j \int \Delta_j f \, \overline{\tilde\Delta_j g}$ for nice $f$ and $g$.
The Cauchy–Schwarz inequality in $j$ and [[Hölder's inequality|JQ9WXC729D]] give $\abs{\int f \bar g} \le \norm{Sf}_p \norm{\tilde S g}_{p'}$, and the upper bound for the square function $\tilde S$ of the $\tilde\Delta_j$ in $L^{p'}$ finishes the proof.

The inequality fails for $p = 1$ and $p = \infty$.
At $p = 1$ the upper bound fails already for an integrable $f$ with $\int f \neq 0$, and the correct substitute replaces $L^1$ by the real Hardy space $H^1$.

## Explanations

### L1 human

In $L^2$, a function split into frequency octaves has a norm that is exactly computable from the norms of the octaves.
The Littlewood–Paley inequality says that in $L^p$, for $1 < p < \infty$, the norm of $f$ can still be read off, up to constants, from the square function.
The square function combines the sizes of the octave pieces at each point.
So the octave pieces of $f$ do not cancel each other too much in $L^p$.

### L2 human

A typical consequence: for any choice of signs $\varepsilon_j \in \set{-1, 1}$, the operator $f \mapsto \sum_j \varepsilon_j \Delta_j f$ is bounded on $L^p$ with a bound independent of the signs.
This is the first step towards the Marcinkiewicz and Hörmander–Mikhlin multiplier theorems, where a multiplier is cut into dyadic pieces and each piece is handled separately.
The inequality is also the starting point for the characterization of Sobolev spaces $W^{k,p}$, Besov spaces and Triebel–Lizorkin spaces through the sizes of the $\Delta_j f$.
On the torus, the analogous statement with sharp dyadic blocks of the Fourier series is the original theorem of Littlewood and Paley.