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Multi-linear operators given by singular multipliers. (English) Zbl 0994.42015

The authors study \(L^{p}\)-boundedness for a class of multilinear operators \(T_{m}\) defined in terms of Fourier multipliers \(m\) by \[ (T_{m}(f_{1},\dots ,f_{n-1}))^{\wedge }(-\xi _{n})=\int \delta (\xi _{1}+\dots +\xi _{n})m(\xi)\hat{f}_{1}(\xi _{1})\cdots \hat{f}_{n-1}(\xi _{n-1}) d\xi _{1}\cdots d\xi _{n-1}. \] where \(\xi =(\xi _{1},\dots ,\xi _{n})\). Expressing \(T_{m}\) this way allows one to obtain bounds for \(T_{m}\) by interpolating from restricted weak-type estimates for the \(n\)-linear form \[ \Lambda _{m}(f_{1},\dots ,f_{n})=\int T_{m}(f_{1},\dots ,f_{n-1})(x)f_{n}(x) dx. \] One says that \(\Lambda \) is of restricted weak-type \(\alpha =(\alpha _{1},\dots ,\alpha _{n})\) provided \[ |\Lambda (f_{1},\dots ,f_{n})|\leq C|E_{1}|^{\alpha _{1}}\cdots |E_{n}|^{\alpha _{n}} \] where \(|f_{i}|\leq 1\) on its support \(E_{i}\). Here one thinks of \(\alpha _{i}=\frac{1}{p_{i}}\). In order to obtain operator bounds \(T_{m}:L^{p_{1}}\times \cdots \times L^{p_{n-1}}\rightarrow L^{p_{n}'}\) that allow \(p_{n}<0\) one must allow \(\alpha _{n}<0\) . Moreover, one only needs \(|f_{n}|\leq 1\) on the complement of an exceptional subset of \(E_{n}\) where the maximal function of \(\chi _{E_{n}}\) is large. Dilation structure imposes the added condition \( \sum \alpha _{i}=1\). Once a restricted weak-type estimate is established, multilinear interpolation is used to pass to \(L^{p}\)-estimates. The restricted weak-type estimates in the current work are based on certain Carleson type tile-norms similar to ones used by the authors in previous work on bilinear singular integrals. Here those Carleson-type estimates are extended to multi-tiles. These are products of tiles – essentially dyadic time-frequency rectangles of unit area – in which each tile in the product shares same time interval. The authors consider symbols \(m\) that can have singularities along a subspace \(\Gamma '\) in \(\Gamma =\{\xi :\xi _{1}+\dots +\xi _{n}=0\}\) having dimension \(k<n/2\) while satisfying symbol estimates \(|\partial _{\xi }^{\alpha }m(\xi)|\leq C_{\alpha }\text{dist} (\xi ,\Gamma ')^{-|\alpha |}\). Suppose for convenience that \(\Gamma '\) is parameterized in terms of \(\xi _{1},\dots ,\xi _{k}\). It is shown that under these conditions \(T_{m}\) continuously maps \(L^{p_{1}}\times \dots \times L^{p_{n-1}}\) into \(L^{p_{n}}\) provided (i) \(1<p_{i}\leq \infty (i=1,\dots ,n-1)\), (ii) \(1/(n-1)<p_{n}'<\infty \), and (iii) \( \frac{1}{p_{1}}+\dots +\frac{1}{p_{n}}=1\). In addition, (iv) \(\frac{1}{p_{i_{1}}} +\dots +\frac{1}{p_{i_{r}}}<\frac{n-2k+r}{2}\) for all choices of \(1\leq i_{1}<\dots <i_{r}\leq n\) and \(1\leq r\leq n\). The range of exponents (i)-(iii) is familiar from work of C. E. Kenig and E. M. Stein [Math. Res. Lett. 6, No. 1, 1-15; 6, No. 3-4, 467 (1999; Zbl 0952.42005)]. The role of the singularity was clarified by J. E. Gilbert and A. R. Nahmod [J. Fourier Anal. Appl. 7, No. 5, 435-467 (2001; preceding review)] who proved the result for the case \(n=3\) and \(k=1\). The present work involves an induction on \(k\). The endpoint case \(k=n/2\) is still open when \(n\) is even, although the authors prove a substitute in which \(L^{p_{i}}\) is replaced by the Wiener algebra in enough coordinates.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
47H60 Multilinear and polynomial operators
47G10 Integral operators
47B25 Linear symmetric and selfadjoint operators (unbounded)
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References:

[1] Calixto P. Calderón, On commutators of singular integrals, Studia Math. 53 (1975), no. 2, 139 – 174. · Zbl 0315.44006
[2] R. R. Coifman and Yves Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975), 315 – 331. · Zbl 0324.44005
[3] R. Coifman and Y. Meyer, Commutateurs d’intégrales singulières et opérateurs multilinéaires, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 3, xi, 177 – 202 (French, with English summary). · Zbl 0368.47031
[4] R. R. Coifman and Y. Meyer, Fourier analysis of multilinear convolutions, Calderón’s theorem, and analysis of Lipschitz curves, Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979) Lecture Notes in Math., vol. 779, Springer, Berlin, 1980, pp. 104 – 122. · Zbl 0427.42006
[5] Ronald R. Coifman and Yves Meyer, Au delà des opérateurs pseudo-différentiels, Astérisque, vol. 57, Société Mathématique de France, Paris, 1978 (French). With an English summary. · Zbl 0483.35082
[6] R. R. Coifman and Yves Meyer, Nonlinear harmonic analysis, operator theory and P.D.E, Beijing lectures in harmonic analysis (Beijing, 1984) Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 3 – 45. · Zbl 0623.47052
[7] Yves Meyer and R. R. Coifman, Ondelettes et opérateurs. III, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1991 (French). Opérateurs multilinéaires. [Multilinear operators]. · Zbl 0745.42012
[8] Charles Fefferman, Pointwise convergence of Fourier series, Ann. of Math. (2) 98 (1973), 551 – 571. · Zbl 0268.42009
[9] Gilbert, J. and Nahmod, A. Boundedness of bilinear operators with non-smooth symbols, Math. Res. Lett. 7 (2000), no. 5-6, 767-778, CMP 2001:07
[10] Gilbert, J. and Nahmod, A. Bilinear Operators with Non-Smooth Symbols I. J. Fourier Anal. and Appl. 7 (2001), 437-469.
[11] Gilbert, J. and Nahmod, A. \(L^p\) - Boundedness of Time-Frequency Paraproducts, to appear in J. Fourier Anal. and Appl. · Zbl 1028.42013
[12] Grafakos, L. and Torres, R. On multilinear singular integrals of Calderon-Zygmund type, to appear in the Proceedings of the El Escorial Conference held in El Escorial, Spain, July 3-7, 2000. · Zbl 1016.42009
[13] Svante Janson, On interpolation of multilinear operators, Function spaces and applications (Lund, 1986) Lecture Notes in Math., vol. 1302, Springer, Berlin, 1988, pp. 290 – 302. · Zbl 0827.46062
[14] Carlos E. Kenig and Elias M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett. 6 (1999), no. 1, 1 – 15. , https://doi.org/10.4310/MRL.1999.v6.n1.a1 Carlos E. Kenig and Elias M. Stein, Erratum to: ”Multilinear estimates and fractional integration”, Math. Res. Lett. 6 (1999), no. 3-4, 467. · Zbl 0952.42005
[15] Michael Lacey and Christoph Thiele, \?^{\?} estimates on the bilinear Hilbert transform for 2&lt;\?&lt;\infty , Ann. of Math. (2) 146 (1997), no. 3, 693 – 724. · Zbl 0914.46034
[16] Michael Lacey and Christoph Thiele, On Calderón’s conjecture, Ann. of Math. (2) 149 (1999), no. 2, 475 – 496. · Zbl 0934.42012
[17] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. · Zbl 0821.42001
[18] Thiele, C. On the Bilinear Hilbert transform, Universität Kiel, Habilitationsschrift [1998]. · Zbl 0915.42011
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