Truncations of multilinear Hankel operators

Bonami, Aline; Grellier, Sandrine; Kacim, Mohammad

doi:10.1090/S0002-9947-07-04185-2

Truncations of multilinear Hankel operators
HTML articles powered by AMS MathViewer

by Aline Bonami, Sandrine Grellier and Mohammad Kacim PDF

Trans. Amer. Math. Soc. 360 (2008), 1377-1390 Request permission

Abstract:

We extend to multilinear Hankel operators the fact that some truncations of bounded Hankel operators are still bounded. We prove and use a continuity property of bilinear Hilbert transforms on products of Lipschitz spaces and Hardy spaces.

References

Aline Bonami and Joaquim Bruna, On truncations of Hankel and Toeplitz operators, Publ. Mat. 43 (1999), no. 1, 235–250. MR 1697523, DOI 10.5565/PUBLMAT_{4}3199_{1}0
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
R. R. Coifman and Yves Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975), 315–331. MR 380244, DOI 10.1090/S0002-9947-1975-0380244-8
Ronald R. Coifman and Guido Weiss, Transference methods in analysis, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 31, American Mathematical Society, Providence, R.I., 1976. MR 0481928
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
Jesus Gasch and John E. Gilbert, Triangularization of Hankel operators and the bilinear Hilbert transform, The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999) Contemp. Math., vol. 247, Amer. Math. Soc., Providence, RI, 1999, pp. 235–248. MR 1738092, DOI 10.1090/conm/247/03804
Loukas Grafakos and Nigel Kalton, Multilinear Calderón-Zygmund operators on Hardy spaces, Collect. Math. 52 (2001), no. 2, 169–179. MR 1852036
Loukas Grafakos and Xiaochun Li, Uniform bounds for the bilinear Hilbert transforms. I, Ann. of Math. (2) 159 (2004), no. 3, 889–933. MR 2113017, DOI 10.4007/annals.2004.159.889
Loukas Grafakos and Rodolfo H. Torres, Multilinear Calderón-Zygmund theory, Adv. Math. 165 (2002), no. 1, 124–164. MR 1880324, DOI 10.1006/aima.2001.2028
Grellier, S. and Kacim, M. Truncation of multilinear Hankel operators, preprint (2003).
Carlos E. Kenig and Elias M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett. 6 (1999), no. 1, 1–15. MR 1682725, DOI 10.4310/MRL.1999.v6.n1.a1
Michael T. Lacey, The bilinear maximal functions map into $L^p$ for $2/3<p\leq 1$, Ann. of Math. (2) 151 (2000), no. 1, 35–57. MR 1745019, DOI 10.2307/121111
Michael Lacey and Christoph Thiele, $L^p$ estimates on the bilinear Hilbert transform for $2<p<\infty$, Ann. of Math. (2) 146 (1997), no. 3, 693–724. MR 1491450, DOI 10.2307/2952458
Michael T. Lacey and Christoph M. Thiele, On Calderón’s conjecture for the bilinear Hilbert transform, Proc. Natl. Acad. Sci. USA 95 (1998), no. 9, 4828–4830. MR 1619285, DOI 10.1073/pnas.95.9.4828
Michael Lacey and Christoph Thiele, On Calderón’s conjecture, Ann. of Math. (2) 149 (1999), no. 2, 475–496. MR 1689336, DOI 10.2307/120971
Camil Muscalu, Terence Tao, and Christoph Thiele, Multi-linear operators given by singular multipliers, J. Amer. Math. Soc. 15 (2002), no. 2, 469–496. MR 1887641, DOI 10.1090/S0894-0347-01-00379-4
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. MR 1232192