Oscillatory integrals with variable Calderón-Zygmund kernel on vanishing generalized Morrey spaces

Abstract

In this paper, the authors investigate the boundedness of the oscillatory singular integrals with variable Calderón-Zygmund kernel on generalized Morrey spaces $M^{p,\varphi}(\mathbb R^n)$ and the vanishing generalized Morrey spaces $VM^{p,\varphi}(\mathbb R^n)$. When $1<p<\infty$ and $(\varphi_1,\varphi_2)$ satisfies some conditions, we show that the oscillatory singular integral operators $T_{\lambda}$ and $T_{\lambda}^{*}$ are bounded from $M^{p,\varphi_1}(\mathbb R^n)$ to $M^{p,\varphi_2}(\mathbb R^n)$ and from $VM^{p,\varphi_1}(\mathbb R^n)$ to $VM^{p,\varphi_2}(\mathbb R^n)$. Meanwhile, the corresponding result for the oscillatory singular integrals with standard Calderón-Zygmund kernel are established.