---
tag: 375ZGE0V2W
kind: statement
role: theorem
name: Marcinkiewicz multiplier theorem
msc: [42A45, 42B15]
uses: [RAGR38Q40B, DPVM34ADZV, SASP14MA1Z]
about: [RAGR38Q40B]
sources:
  - url: https://en.wikipedia.org/wiki/Multiplier_(Fourier_analysis)
---

## Statement

Let $m \colon \R \to \C$ be a bounded function, with $\abs{m(\xi)} \le A$ for all $\xi \in \R$.
For each integer $j$, let $I_j = [2^j, 2^{j+1}]$ and $I_j' = [-2^{j+1}, -2^j]$.
Assume that $m$ has bounded variation on each $I_j$ and each $I_j'$, with total variation at most $A$ on each of these intervals.
Then $m$ is an [[$L^p$ multiplier|RAGR38Q40B]] for every $1 < p < \infty$.
Let $\hat f$ be the [[Fourier transform|DPVM34ADZV]] of $f$.
There is a constant $C_p$, depending only on $p$, with
$$\norm{(m \hat f)^\vee}_{L^p(\R)} \le C_p A \norm{f}_{L^p(\R)}$$
for every Schwartz function $f$.

## Remarks

The proof uses Littlewood–Paley theory.
The operators that cut the frequencies to the dyadic pieces $I_j$ and $I_j'$ split $f$ into pieces whose square function has $L^p$ norm comparable to that of $f$.
On each dyadic piece, a function of bounded variation is an average of indicators of intervals.
The multiplier of an interval is built from Hilbert transforms, which are bounded on $L^p$.
A vector-valued bound for these interval operators then gives the theorem.

Example: $m(\xi) = \abs{\xi}^{it}$ for a real $t \ne 0$ has modulus $1$, and on $[2^j, 2^{j+1}]$ its variation is $\int_{2^j}^{2^{j+1}} \abs{t}/\xi \, d\xi = \abs{t} \log 2$.
So the imaginary powers of $\abs{\xi}$ are $L^p$ multipliers for $1 < p < \infty$.

The conclusion fails at $p = 1$ and $p = \infty$: the multiplier $m = \mathbf{1}_{[0, \infty)}$ satisfies the hypotheses but gives an operator built from the Hilbert transform, which is unbounded on $L^1$.
There is also a version on $\R^n$, in which the condition is imposed on the dyadic rectangles, using mixed derivatives of $m$.

## Explanations

### L1 human

Cut the frequency line into blocks $[2^j, 2^{j+1}]$ that double in length, and the same on the negative side.
The theorem says that a bounded multiplier that does not wiggle much within each block preserves $L^p$ size for $1 < p < \infty$.
It may jump between blocks, and it may have infinitely many jumps in total.

### L2 human

This criterion needs no smoothness: only bounded variation on each dyadic block.
It is weaker in hypotheses than the [[Hörmander–Mikhlin multiplier theorem|F5V7CXW30E]] in one variable, where $\abs{m'(\xi)} \le A/\abs{\xi}$ implies a variation of at most $A \log 2$ on each block.
Its proof is a model use of Littlewood–Paley theory together with the $L^p$ boundedness of the Hilbert transform.