Bilinear Pseudo-Differential Operator and Its Commutator on Generalized Fractional Weighted Morrey Spaces (Analysis in Theory and Applications, 2024)
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 $L^{p,η,φ} (ω).$ Under assumption that a weight satisfies a certain condition, the authors prove that $T_σ$ isbounded from products of spaces $L^{p_1,η_1,φ}(ω_1)×L^{p_2,η_2,φ}(ω_2)$ into spaces $L^{p,η,φ} (\vec{ω}),$ 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 productsof generalized fractional Morrey spaces $L^{p_1,η_1,φ} (\mathbb{R}^n)×L^{p_2,η_2,φ} (\mathbb{R}^n)$ into $L^{p,η,φ}(\mathbb{R}^n).$ Ascorollaries, the boundedness of the $T_σ$ and $[b_1, b_2, T_σ]$ on generalized weighted Morreyspaces $L^{p,φ} (ω)$ and on generalized Morrey spaces $L^{p,φ}(\mathbb{R}^n)$ is also obtained.