Coifman–Rochberg–Weiss commutator theorem

Remarks

One proof of part 1 uses a conjugation trick. For complex $z$, the operator $T_z f = e^{zb} T(e^{-zb} f)$ satisfies $\frac{d}{dz} T_z \big\vert_{z = 0} = [b, T]$. By the John–Nirenberg inequality, $e^{p \, \mathrm{Re}(z) b}$ is a Muckenhoupt $A_p$ weight when $\abs{z} \norm{b}_{\mathrm{BMO}}$ is small. So, by weighted theory, $T_z$ is bounded on $L^p$ uniformly for small $\abs{z}$. The Cauchy integral formula for the derivative then gives the bound on $[b, T]$.

For $n = 1$ the only Riesz transform is the Hilbert transform $H$, and the theorem says that $[b, H]$ is bounded on $L^p(\R)$ exactly when $b \in \mathrm{BMO}(\R)$. Example: $b(x) = \log \abs{x}$ is unbounded, so the multiplication operator by $b$ is unbounded on $L^p$, yet $[b, H]$ is bounded because $b \in \mathrm{BMO}$.

Going deeper

For a student

The theorem gives a new characterization of BMO through operators, complementing its definition by mean oscillations and Fefferman's duality with $H^1$. By duality it is equivalent to a factorization theorem for $H^1$: every $f \in H^1(\R^n)$ can be written as $\sum_k \sum_j \left( g_{k,j} R_j h_{k,j} + h_{k,j} R_j g_{k,j} \right)$ with $\sum_k \sum_j \norm{g_{k,j}}_{p'} \norm{h_{k,j}}_p < \infty$. Commutators with BMO functions appear in the study of div-curl lemmas in compensated compactness and of elliptic equations with VMO coefficients. The commutator $[b, T]$ is compact on $L^p$ exactly when $b$ is in the closure of smooth compactly supported functions in BMO, a result of Uchiyama.