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Multilinear oscillatory integrals with Calderón-Zygmund kernel. (English) Zbl 0954.42008

The author establishes an \(L^p\)-boundedness criterion for a class of multilinear oscillatory singular integrals with standard Calderón-Zygmund kernels. Let \[ R_m(A; x,y)= A(x)- \sum_{|\alpha|< m} {D^\alpha A(y)\over \alpha!} (x- y)^\alpha, \]
\[ T_{A_1,A_2}f(x)=\text{ p.v. }\int_{\mathbb{R}^n} e^{iP(x,y)}{K(x,y)\over |x-y|^{M-1}} \prod^2_{j= 1} R_{m_j}(A_j; x,y) f(y) dy, \] where \(M= m_1+ m_2\); \(P(x,y)\) is a real-valued polynomial in \(x\) and \(y\); \(A_1(x)\) is a function that satisfies \(D^\alpha A_1\in \text{BMO}(\mathbb{R}^n)\) for all multi-indices \(\alpha\) with \(|\alpha|= m_1- 1\); \(A_2\) has derivatives of order \(m_2\) in \(L^s(\mathbb{R}^n)\), \(1<s\leq \infty\), and \(K(x,y)\) is a standard Calderón-Zygmund kernel. The truncated operator \(S_{A_1,A_2}f(x)\) is defined by \[ S_{A_1,A_2}f(x)= \text{ p.v. } \int_{|x-y|< 1}K(x,y)|x-y|^{- M+1} \prod^2_{j= 1} R_{m_j}(A_j; x,y) f(y) dy. \] The author proved that if \(P(x,y)\) satisfies certain conditions, then, for \(1/r= 1/p+ 1/s\), \(1< r\), \(p<\infty\), \[ \|T_{A_1,A_2}f\|_{L^r}\leq C_1\Biggl( \sum_{|\alpha|= m_1- 1}\|D^\alpha A_1\|_{\text{BMO}}\Biggr) \Biggl(\sum_{|\beta|= m_2}\|D^\beta A_2\|_{L^s}\Biggr) \|f\|_{L^p} \] if and only if \[ \|S_{A_1,A_2}f\|_{L^r}\leq C_2 \Biggl( \sum_{|\alpha|= m_1- 1}\|D^\alpha A_1\|_{\text{BMO}}\Biggr) \Biggl( \sum_{|\beta|= m_2}\|D^\beta A_2\|_{L^s}\Biggr) \|f\|_{L^p}, \] where \(C_1\), \(C_2\) are constants depending only on \(n\), \(p\), and \(\deg(P)\).
This result is also true if \(r= p\), \(A_2\) has derivatives of order \(m_2\) in \(\text{BMO}(\mathbb{R}^n)\) and \(\sum_{|\beta|= m_2}\|D^\beta A_2\|_{L^s}\) is replaced by \(\sum_{|\beta|= m_2}\|D^\beta A_2\|_{\text{BMO}}\).

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B30 \(H^p\)-spaces
42B35 Function spaces arising in harmonic analysis
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