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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.
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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