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Multilinear Singular Integral Forms of Christ-Journé Type
About this Title
Andreas Seeger, Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706, Charles K. Smart, Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637 and Brian Street, Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, Wisconsin 53706
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 257, Number 1231
ISBNs: 978-1-4704-3437-3 (print); 978-1-4704-4945-2 (online)
DOI: https://doi.org/10.1090/memo/1231
Published electronically: January 2, 2019
Keywords: Multilinear singular integral forms,
Christ-Journé operators,
mixing flows
MSC: Primary 42B20
Table of Contents
Chapters
- 1. Introduction
- 2. Statements of the main results
- 3. Kernels
- 4. Adjoints
- 5. Outline of the proof of boundedness
- 6. Some auxiliary operators
- 7. Basic $L^2$ estimates
- 8. Some results from Calderón-Zygmund theory
- 9. Almost orthogonality
- 10. Boundedness of Multilinear Singular Forms
- 11. Interpolation
Abstract
We introduce a class of multilinear singular integral forms \[ \Lambda :L^{p_1}(\mathbb {R}^d)\times \cdots \times L^{p_{n+2}}(\mathbb {R}^d)\rightarrow \mathbb C\] which generalize the Christ-Journé multilinear forms; here $\sum _{j=1}^{n+2} p_j^{-1}=1$, $p_j\in (1,\infty ]$. The research is partially motivated by an approach to Bressan’s problem on incompressible mixing flows. A key aspect of the theory is that the class of operators is closed under adjoints (i.e. the class of multilinear forms is closed under permutations of the entries). This, together with an interpolation, allows us to reduce the $L^{p_1}\times \cdots \times L^{p_{n+2}}$ boundedness to $L^\infty \times \cdots \times L^\infty \times L^p\times L^{p’}$ boundedness. We obtain estimates of the form \begin{equation*} |\Lambda (f_1,\ldots , f_{n+2})| \leq C n^2 \log ^3(2+n) \prod _{j=1}^{n+2} \| f_j\|_{L^{p_j}}, \end{equation*} where the constant $C$ does not depend on $n$.- J. Bergh, J. Löfström, Interpolation spaces. Springer-Verlag, Berlin, Heidelberg, New York, 1976.
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