Bilinear Calderón-Zygmund operator and its commutator on some variable exponent spaces of homogeneous type

Authors : Guanghui Lu

Pages : 433-456

Doi:10.15672/hujms.1195476

View : 138 | Download : 198

Publication Date : 2024-04-23

Article Type : Research Paper

Abstract :Let $(X,d,\\mu)$ be a space of homogeneous type in the sense of Coifman and and Weiss. In this setting, the author proves that a bilinear Calderon-Zygmund operator is bounded from the product of variable exponent Lebesgue spaces $L^{p_{1}(\\cdot)}(X)\\times L^{p_{2}(\\cdot)}(X)$ into spaces $L^{p(\\cdot)}(X)$, and it is bounded from the product of variable exponent generalized Morrey spaces $\\mathcal{L}^{p_{1}(\\cdot),\\varphi_{1}}(X)\\times \\mathcal{L}^{p_{2}(\\cdot),\\varphi_{2}}(X)$ into spaces $\\mathcal{L}^{p(\\cdot),\\varphi}(X)$, where the Lebesgue measure functions $\\varphi(\\cdot,\\cdot), \\varphi_{1}(\\cdot,\\cdot)$ and $\\varphi_{2}(\\cdot,\\cdot)$ satisfy $\\varphi_{1}\\times\\varphi_{2}=\\varphi$, and $\\frac{1}{p(\\cdot)}=\\frac{1}{p_{1}(\\cdot)}+\\frac{1}{p_{2}(\\cdot)}$. Furthermore, by establishing sharp maximal estimate for the commutator $[b_{1},b_{2},BT]$ generated by $b_{1}, b_{2}\\in\\mathrm{BMO}(X)$ and $BT$, the author shows that the $[b_{1},b_{2},BT]$ is bounded from the product of spaces $L^{p_{1}(\\cdot)}(X)\\times L^{p_{2}(\\cdot)}(X)$ into spaces $L^{p(\\cdot)}(X)$, and it is also bounded from product of spaces $\\mathcal{L}^{p_{1}(\\cdot),\\varphi_{1}}(X)\\times \\mathcal{L}^{p_{2}(\\cdot),\\varphi_{2}}(X)$ into spaces $L^{p(\\cdot),\\varphi}(X)$.
Keywords : space of homogeneous type, bilinear Calderón Zygmund operator, commutator, space BMO X, variable exponent generalized Morrey space

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