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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 \(\|HB-BH\|\leq C(1+\log n)\|B\|_{\text{BMO}}\).

42A50 Conjugate functions, conjugate series, singular integrals
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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