Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T15:00:10.586Z Has data issue: false hasContentIssue false
  • English
  • Français

Transference on certain multilinear multiplier operators

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  09 April 2009

Dashan Fan  and
Shuichi Sato
Dashan Fan
Affiliation:
Department of Mathematics Huazhong University of Science and Technology and Department of Mathematics University of Wisconsin-MilwaukeeMilwaukee, WI 53201USA e-mail: fan@csd.uwm.edu
Shuichi Sato
Affiliation:
Department of Mathematics Kanazawa UniversityKanazawa 920-11Japan e-mail: shuichi@kenroku.kanazawa-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study DeLeeuw type theorems for certain multilinear operators on the Lebesgue spaces and on the Hardy spaces. As applications, on the torus we obtain an analog of Lacey—Thiele's theorem on the bilinear Hilbert transform, as well as analogies of some recent theorems on multilinear singular integrals by Kenig—Stein and by Grafakos—Torres.

MSC classification

Secondary: 42B15: Multipliers 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 70 , Issue 1 , February 2001 , pp. 37 - 55
Copyright
Copyright © Australian Mathematical Society 2001

References

[AC]Auscher, P. and Carro, M. J., ‘On relations between operators on Rn, Tn and Zn’, Studia Math. 101 (1992), 165–182.CrossRefGoogle Scholar
[BF]Blank, B. and Fan, D., ‘S-functions and gλ-functions on compact Lie groups’, J. Austral. Math. Soc. (Series A) 61 (1996), 327–344.CrossRefGoogle Scholar
[CG]Coifman, R. and Grafakos, L., ‘Hardy space estimates for multilinear operators (I)’, Rev. Mat. Iberoamericana 8 (1992), 45–62.Google Scholar
[CM 1]Coifman, R. and Meyer, Y., ‘On commutators of singular integrals and bilinear singular integrals’, Trans. Amer Math. Soc. 212 (1975), 315–331.CrossRefGoogle Scholar
[CM2]Coifman, R. and Meyer, Y., ‘Commutateurs d'integrales singulières et opérateurs multilinéaires’, Ann. of Inst. Fourier (Grenoble) 28 (1978), 177–202.Google Scholar
[CM3]Coifman, R. and Meyer, Y., Au-delà des opérateurs pseudo-differentiels, Astérisque 57 (Société Mathématique de France, Paris, 1978).Google Scholar
[CM4]Coifman, R. and Meyer, Y., ‘Non-linear harmonic analysis, operator theory and P.D.E.’, Beijing Lectures in Analysis, Ann. of Math. Stud. 112 (Princeton Univ. Press, Princeton, 1986), pp. 3–46.Google Scholar
[F]Fan, D., ‘Multipliers on certain function spaces’, Rend. Circ. Mat. Palermo (2) 43 (1994), 449–463.CrossRefGoogle Scholar
[FS]Folland, G. and Stein, E. M., Hardy spaces on homogeneous groups (Princeton Univ. Press, Princeton, 1982).Google Scholar
[G]Grafakos, L., ‘Hardy space estimates for multilinear operators (II)’, Rev. Mat. Iberoamericana 8 (1992), 69–92.Google Scholar
[GK]Grafakos, L. and Kalton, N., ‘Some remarks on multilinear maps and interpolation’, to appear.Google Scholar
[GT]Grafakos, L. and Torres, R., ‘Multilinear Calderón-Zygmund theory’, to appear.Google Scholar
[GW]Grafakos, L. and Weiss, G., ‘Transference of multilinear operators’, Illinois J. Math. 40 (1996), 344–351.CrossRefGoogle Scholar
[K]Kaneko, M., ‘Boundedness of some operators composed by Fourier multipliers’, Tohoku Math. J. 35 (1983), 267–288.CrossRefGoogle Scholar
[KaS]Kaneko, M. and Sato, E., ‘Notes on transference of continuity from maximal Fourier multiplier operators on Rn to those on Tn’, Interdiscip. Inform. Sci. 4 (1998), 97–107.Google Scholar
[KeS]Kenig, C. and Stein, E. M., ‘Multilinear estimates and fractional integration’, Math. Research Lett. 6 (1999), 1–15.CrossRefGoogle Scholar
[KT]Kenig, C. and Thomas, P., ‘Maximal operators defined by Fourier multiplier’, Studia Math. 68 (1980), 79–83.CrossRefGoogle Scholar
[L]DeLeeuw, K., ‘On L p multiplier’, Ann. of Math. 91 (1965), 364–379.CrossRefGoogle Scholar
[La]Lacey, M., ‘On the bilinear Hubert transform’, in: Proceedings of International Congress of Mathematicians, vol. II, (Berlin, 1998),Google Scholar
Doc. Math. 1998, Extra vol. II, pp. 647–656.Google Scholar
[LT]Lacey, M. and Thiele, C., ‘L p estimates on the bilinear Hilbert transform’, Ann. of Math. (2) 146 (1997), 693–724.CrossRefGoogle Scholar
[LL]Liu, Z. and Lu, S., ‘Transference and restriction of maximal multiplier operators on Hardy spaces’, Studia Math. 105 (1993), 121–134.CrossRefGoogle Scholar
[S]Stein, E. M., Harmonic analysis, real-variable methods, orthogonality, and oscillatory integrals (Princeton Univ. Press, Princeton, 1993).Google Scholar
[SW]Stein, E. M. and Weiss, G., Introduction to Fourier analysis on Euclidean spaces (Princeton Univ. Press, Princeton, 1971).Google Scholar
[T]Torres, R., ‘Spaces of sequence, sampling theorems and functions of exponential type’, Studia Math. 100 (1991), 51–74.CrossRefGoogle Scholar