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Oscillatory integrals with variable Calderón-Zygmund kernel on vanishing generalized Morrey spaces. (English) Zbl 1439.42017
Summary: 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.
##### MSC:
 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory 42B35 Function spaces arising in harmonic analysis
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