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Uniform boundedness of oscillatory singular integrals on Hardy spaces. (English) Zbl 0998.42009

For \(n\in \mathbb N\), let \(\varphi\in C^{\infty}_0({\mathbb R}^n)\) and \(\Phi\in C^{\infty}({\mathbb R}^n)\) be a real-valued function satisfying \(\nabla\Phi(0)=0\). Let \(K\) be a Calderón-Zygmund kernel on \({\mathbb R}^n\), i.e., \(|K(x)|+|x||\nabla K(x)|\leq A |x|^{-n}\) and \(\int_{a<|x|<b} K(x) dx =0\) for \(b>a>0\). Then the authors study the uniform boundedness on the Hardy space \(H^1({\mathbb R}^n)\) of the localized oscillatory singular integral operator \(T_{\lambda}\) on \({\mathbb R}^n\) defined by \[ (T_{\lambda} f)(x)=\text{p.v. }\int_{{\mathbb R}^n} e^{i\lambda\Phi (x-y)} K(x-y) \varphi(x-y) f(y) dy. \] Such oscillatory singular integral operators have arisen in many problems in harmonic analysis and related areas and have been studied extensively. The working on the boundedness on \(L^p({\mathbb R}^n)\) and \(H^p({\mathbb R}^n)\) of such operators has previously been done by D. H. Phong and E. M. Stein [Acta Math. 157, 99-157 (1986; Zbl 0622.42011)], F. Ricci and E. M. Stein [J. Funct. Anal. 73, 179-194 (1987; Zbl 0622.42010)], Y. Pan [J. Funct. Anal. 100, No. 1, 207-220 (1991; Zbl 0735.45010) and Indiana Univ. Math. J. 41, No. 1, 279-293 (1992; Zbl 0779.42008)], and A. Carbery, M. Christ and J. Wright [J. Am. Math. Soc. 12, No. 4, 981-1015 (1999; Zbl 0938.42008)].
Inspired by the work of Carbery, Christ and Wright, the present authors establish the following result: If at least one coordinate of a multi-index \(\alpha\) with \(|\alpha|\geq 3\) is strictly greater than \(1\) and there exist \(\delta>0\), \(A>0\) such that \[ \max_{|\beta|=|\alpha|}\sup_{|x|\leq s} |{\mathcal D}^{\beta}\Phi (x)|\leq A\inf_{s\leq |x|\leq\delta} |{\mathcal D}^{\alpha}\Phi (x)| \] holds for all \(s\in (0,\delta)\), then there exists a constant \(C>0\) such that \[ \|T_{\lambda} f\|_{H^1({\mathbb R}^n)}\leq C \|f\|_{H^1({\mathbb R}^n)} \] for \(f\in H^1({\mathbb R}^n)\) and \(\lambda\in\mathbb R\).

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B30 \(H^p\)-spaces
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