Higher-order commutators of parameterized Littlewood-Paley operators on two weight Herz spaces with variable exponents
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In this article, we consider the boundedness for a class of parameterized Littlewood-Paley integrals and their commutators. More precisely, let $\Omega\in L^{2}\left(S^{n-1}\right)$ be a homogeneous function of degree zero. We prove that parameterized Littlewood-Paley area integral $\mu_{\Omega,S}^{\rho}$, $g_{\lambda}^{*}$ function $\mu_{\Omega,\lambda}^{*,\rho}$ are bounded on two weighted Herz spaces with variable exponents. It is worth noting that above operators in two weighted Herz spaces with variable exponents are more complex than operators themselves. Moreover, let $b$ be a BMO function, the boundedness of commutators generated by $b$ and parameterized Littlewood-Paley operators will also be showed.
Abstract
In this article, we consider the boundedness for a class of parameterized Littlewood-Paley integrals and their commutators. More precisely, let $\Omega\in L^{2}\left(S^{n-1}\right)$ be a homogeneous function of degree zero. We prove that parameterized Littlewood-Paley area integral $\mu_{\Omega,S}^{\rho}$, $g_{\lambda}^{*}$ function $\mu_{\Omega,\lambda}^{*,\rho}$ are bounded on two weighted Herz spaces with variable exponents. It is worth noting that above operators in two weighted Herz spaces with variable exponents are more complex than operators themselves. Moreover, let $b$ be a BMO function, the boundedness of commutators generated by $b$ and parameterized Littlewood-Paley operators will also be showed.