Estimates for the commutator of bilinear Fourier multiplier

Hu, Guoen ; Yi, Wentan

Czechoslovak Mathematical Journal, Tome 63 (2013) no. 4, pp. 1113-1134.

Résumé

Let $b_1, b_2 \in {\rm BMO}(\mathbb {R}^n)$ and $T_{\sigma }$ be a bilinear Fourier multiplier operator with associated multiplier $\sigma $ satisfying the Sobolev regularity that $\sup _{\kappa \in \mathbb {Z}} \|\sigma _{\kappa }\| _{W^{s_1,s_2}(\mathbb {R}^{2n})}\infty $ for some $s_1,s_2\in (n/2,n]$. In this paper, the behavior on $L^{p_1}(\mathbb {R}^n)\times L^{p_2}(\mathbb {R}^n)$ $(p_1,p_2\in (1,\infty ))$, on $H^1(\mathbb {R}^n)\times L^{p_2}(\mathbb {R}^n)$ $(p_2\in [2,\infty ))$, and on $H^1(\mathbb {R}^n)\times H^1(\mathbb {R}^n)$, is considered for the commutator $T_{{\sigma }, \vec {b}} $ defined by $$ \begin {aligned} T_{\sigma ,\vec {b}} (f_1,f_2) (x)=(x)T_{\sigma }(f_1, f_2)(x)-T_{\sigma }(b_1f_1, f_2)(x) + b_2(x)T_{\sigma }(f_1, f_2)(x)-T_{\sigma }(f_1, b_2f_2)(x) . \end {aligned} $$ By kernel estimates of the bilinear Fourier multiplier operators and employing some techniques in the theory of bilinear singular integral operators, it is proved that these mapping properties are very similar to those of the bilinear Fourier multiplier operator which were established by Miyachi and Tomita.

Détail

DOI : 10.1007/s10587-013-0074-5

Classification : 42B15
Keywords: bilinear Fourier multiplier operator; commutator; Hardy space