On the oscillatory operators of Calder\'on type commutators on Morrey spaces
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This paper is devoted to studying the boundedness and compactness of oscillatory operators associated with Calder\'on type commutators on Morrey spaces, which are defined by $$T_{\Omega,A}^{P}f(x):={\rm p.v.}\int_{\mathbb{R}^n}e^{iP(x,y)}\frac{\Omega(x-y)}{|x-y|^{n+m-1}}R_m(A;x,y)f(y)dy,$$ where $P(x,y)$ is a real-valued polynomial on $\mathbb{R}^n\times\mathbb{R}^n$, $\Omega$ is homogeneous of degree zero on $\mathbb{R}^n$ and has vanishing moments of order $m-1$ ($m\ge 1$) over the unit sphere $\mathbb S^{n-1}$, $R_m(A;x,y)$ denotes the $m$-th Taylor series remainder of $A$ at $x$ about $y$, $A$ has derivatives of order $m-1$ in ${\rm BMO}(\mathbb{R}^n)$. The authors establish the boundedness of $T_{\Omega,A}^{P}$ with rough kernels on Morrey spaces, and give a criterion on the compactness of $T_{\Omega,A}^{P}$ on Morrey spaces. As application, the compactness of $T_{\Omega,A}^{P}$ with rough kernels on Morrey spaces is obtained.
Abstract
This paper is devoted to studying the boundedness and compactness of oscillatory operators associated with Calder\'on type commutators on Morrey spaces, which are defined by $$T_{\Omega,A}^{P}f(x):={\rm p.v.}\int_{\mathbb{R}^n}e^{iP(x,y)}\frac{\Omega(x-y)}{|x-y|^{n+m-1}}R_m(A;x,y)f(y)dy,$$ where $P(x,y)$ is a real-valued polynomial on $\mathbb{R}^n\times\mathbb{R}^n$, $\Omega$ is homogeneous of degree zero on $\mathbb{R}^n$ and has vanishing moments of order $m-1$ ($m\ge 1$) over the unit sphere $\mathbb S^{n-1}$, $R_m(A;x,y)$ denotes the $m$-th Taylor series remainder of $A$ at $x$ about $y$, $A$ has derivatives of order $m-1$ in ${\rm BMO}(\mathbb{R}^n)$. The authors establish the boundedness of $T_{\Omega,A}^{P}$ with rough kernels on Morrey spaces, and give a criterion on the compactness of $T_{\Omega,A}^{P}$ on Morrey spaces. As application, the compactness of $T_{\Omega,A}^{P}$ with rough kernels on Morrey spaces is obtained.