6533b858fe1ef96bd12b618b

RESEARCH PRODUCT

Commutators of linear and bilinear Hilbert transforms

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Let α ∈ R \alpha \in \mathbb {R} , and let H α ( f , g ) ( x ) = 1 π p . v . ∫ f ( x − t ) g ( x − α t ) d t t H_\alpha (f,g)(x)=\frac {1}{\pi } p.v. \int f(x-t)g(x-\alpha t)\frac {dt}{t} and H f ( x ) = 1 π p . v . ∫ f ( x − t ) d t t Hf(x)= \frac {1}{\pi } p.v.\int f(x-t)\frac {dt}{t} denote the bilinear and linear Hilbert transforms, respectively. It is proved that, for 1 > p > ∞ 1>p>\infty and α 1 ≠ α 2 \alpha _1\ne \alpha _2 , H α 1 − H α 2 H_{\alpha _1}-H_{\alpha _2} maps L p × B M O L^p\times BMO into L p L^{p} and it maps B M O × L p BMO \times L^p into L p L^{p} if and only if sign ⁡ ( α 1 ) = sign ⁡ ( α 2 ) \operatorname {sign}(\alpha _1)=\operatorname {sign}(\alpha _2) . It is also shown that, for α ≤ 1 \alpha \le 1 the commutator [ H α , f , H ] [H_{\alpha ,f},H] is bounded on L p L^p for 1 > p > ∞ 1>p>\infty if and only if f ∈ B M O f\in BMO , where H α , f ( g ) = H α ( f , g ) H_{\alpha ,f}(g)=H_\alpha (f,g) .