TY - JOUR
T1 - Bilinear Pseudo-Differential Operator and Its Commutator on Generalized Fractional Weighted Morrey Spaces
AU - Lu , Guanghui
AU - Tao , Shuangping
JO - Analysis in Theory and Applications
VL - 1
SP - 92
EP - 110
PY - 2024
DA - 2024/04
SN - 40
DO - http://doi.org/10.4208/ata.OA-2021-0016
UR - https://global-sci.org/intro/article_detail/ata/23021.html
KW - Generalized fractional weighted Morrey space, bilinear pseudo-differential operator, commutator, space BMO$(\mathbb{R}^n).$
AB - <p style="text-align: justify;">The aim of this paper is to establish the boundedness of bilinear pseudo-differential operator $T_σ$ and its commutator $[b_1, b_2, T_σ]$ generated by $T_σ$ and $b_1, b_2∈ {\rm BMO}(\mathbb{R}^n)$ on generalized fractional weighted Morrey spaces $L^{p,η,\varphi} (ω).$ Under assumption that a weight satisfies a certain condition, the authors prove that $T_σ$ is
bounded from products of spaces $L^{p_1,η_1,\varphi}(ω_1)×L^{p_2,η_2,\varphi}(ω_2)$ into spaces $L^{p,η,\varphi} (\vec{ω}),$ where $\vec{ω}= (ω_1, ω_2) ∈ A_{\vec{P}},$ $\vec{P} = (p_1, p_2),$ $η = η_1 + η_2$ and $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ with $p_1, p_2 ∈
(1, ∞).$ Furthermore, the authors show that the $[b_1, b_2, T_σ]$ is bounded from products
of generalized fractional Morrey spaces $L^{p_1
,η_1
,\varphi} (\mathbb{R}^n)×L^{p_2,η_2,\varphi} (\mathbb{R}^n)$ into $L^{p,η,\varphi}(\mathbb{R}^n).$ As
corollaries, the boundedness of the $T_σ$ and $[b_1, b_2, T_σ]$ on generalized weighted Morrey
spaces $L^{p,\varphi} (ω)$ and on generalized Morrey spaces $L^{p,\varphi}(\mathbb{R}^n)$ is also obtained.</p>