Marcinkiewicz multiplier theorem

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 $m$.

Going deeper

For a student

This criterion needs no smoothness: only bounded variation on each dyadic block. It is weaker in hypotheses than the Hörmander–Mikhlin multiplier theorem 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.