Estimates for the commutator of bilinear Fourier multiplier

Hu, Guoen, and Yi, Wentan. "Estimates for the commutator of bilinear Fourier multiplier." Czechoslovak Mathematical Journal 63.4 (2013): 1113-1134. <http://eudml.org/doc/260818>.

@article{Hu2013,
abstract = {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\}\} \Vert \sigma _\{\kappa \}\Vert _\{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 \[ \begin\{aligned\} T\_\{\sigma ,\vec\{b\}\} (f\_1,f\_2) (x)=&b\_1(x)T\_\{\sigma \}(f\_1, f\_2)(x)-T\_\{\sigma \}(b\_1f\_1, f\_2)(x) &+ b\_2(x)T\_\{\sigma \}(f\_1, f\_2)(x)-T\_\{\sigma \}(f\_1, b\_2f\_2)(x) . \end\{aligned\} \] 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.},
author = {Hu, Guoen, Yi, Wentan},
journal = {Czechoslovak Mathematical Journal},
keywords = {bilinear Fourier multiplier operator; commutator; Hardy space; bilinear Fourier multiplier operator; commutator; Hardy space},
language = {eng},
number = {4},
pages = {1113-1134},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Estimates for the commutator of bilinear Fourier multiplier},
url = {http://eudml.org/doc/260818},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Hu, Guoen
AU - Yi, Wentan
TI - Estimates for the commutator of bilinear Fourier multiplier
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 4
SP - 1113
EP - 1134
AB - 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}} \Vert \sigma _{\kappa }\Vert _{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 \[ \begin{aligned} T_{\sigma ,\vec{b}} (f_1,f_2) (x)=&b_1(x)T_{\sigma }(f_1, f_2)(x)-T_{\sigma }(b_1f_1, f_2)(x) &+ b_2(x)T_{\sigma }(f_1, f_2)(x)-T_{\sigma }(f_1, b_2f_2)(x) . \end{aligned} \] 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.
LA - eng
KW - bilinear Fourier multiplier operator; commutator; Hardy space; bilinear Fourier multiplier operator; commutator; Hardy space
UR - http://eudml.org/doc/260818
ER -