Commutators for Fourier multipliers on Besov Spaces

The mapping properties of commutators [T,M ] = TM −MT , for operators between function spaces, and their various generalizations play an important role in harmonic analysis, PDE, interpolation theory and other related areas. A typical situation arises when M = Mb is the pointwise multiplication by a function b and T is a Calderón–Zygmund operator on R. Then well– known results of A.P. Calderón [4] and Coifman, Rochberg and Weiss [8] state, respectively, that the commutator is bounded from L2 into W 1,2 if b is a Lipschitz function, and from Lp into Lp if 1 < p < ∞ and b ∈ BMO. Recently, its boundedness between some other function spaces, including Besov–Lipschitz and Triebel–Lizorkin spaces, has been extensively studied (see [13] and the references therein). A related situation appears when M = Tμ is the Fourier multiplier with symbol μ, i.e. T̂μf = μf̂ , where f̂ is the Fourier transform of f . It was proved in [7] that, for Besov spaces of periodic functions Bσ,q p (T), the commutator [T, Tμ] : B p (T)→ B p (T)