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Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol. (English. Abridged French version) Zbl 0991.42003
An $$(n\times n)$$-matrix-valued function $$B$$ is said to belong to BMO if $\sup_J \langle\|Be-\langle B\rangle _J e\|^2 \rangle _J \leq a^2 \|e\|^2$ and the same is true with $$B$$ replaced by $$B^*$$. Here $$J$$ is an arbitrary interval in $${\mathbb R}$$, and $$\langle\cdot\rangle _J$$ stands for the average over $$J$$. The smallest possible $$a$$ is the BMO-norm of $$B$$. Let $$H$$ be the Hilbert transform.
It is proved that the commutator $$HB-BH$$ acts from $$L^2 ({\mathbb R},dx;{\mathbb C}^n)$$ into itself for $$B\in \text{BMO}$$ and that $$\|HB-BH\|\leq C(1+\log n)\|B\|_{\text{BMO}}$$.

##### MSC:
 42A50 Conjugate functions, conjugate series, singular integrals 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 44A15 Special integral transforms (Legendre, Hilbert, etc.)
##### Keywords:
Hilbert transform; Hankel operator; BMO; commutator
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