Hörmander–Mikhlin multiplier theorem

Remarks

The condition with $\alpha = 0$ says that $m$ is bounded, so the operator is bounded on $L^2$ by the Plancherel theorem. The derivative conditions make the inverse Fourier transform $K$ of $m$ a kernel that satisfies the Hörmander condition $\int_{\abs{x} > 2\abs{y}} \abs{K(x - y) - K(x)} \, dx \le CA$. Then the Calderón–Zygmund theorem gives the weak type $(1, 1)$ bound, interpolation gives $1 < p \le 2$, and duality gives $2 \le p < \infty$. Hörmander weakened the pointwise derivative bounds to bounds on the $L^2$ averages of the derivatives over dyadic shells.

Example: in $\R^n$, $m(\xi) = \xi_j / \abs{\xi}$ is smooth away from $0$ and homogeneous of degree $0$. So $\partial^\alpha m$ is homogeneous of degree $-\abs{\alpha}$ and bounded on the unit sphere, which gives the hypothesis. Hence the Riesz transforms are bounded on $L^p(\R^n)$ for $1 < p < \infty$.

Some smoothness is needed. The indicator of a ball is bounded but not smooth. By Fefferman's ball multiplier theorem, it is not an $L^p$ multiplier for $p \ne 2$ when $n \ge 2$.

Going deeper

For a student

This is the most used multiplier criterion in PDE. It shows at once that the Riesz transforms, the operators $\partial_i \partial_j \Delta^{-1}$, and the imaginary powers $(-\Delta)^{it}$ are bounded on $L^p(\R^n)$ for $1 < p < \infty$. It sits between the Marcinkiewicz multiplier theorem, which uses dyadic rectangles, and the general Calderón–Zygmund theory, which supplies its proof.