Hostname: page-component-7479d7b7d-68ccn Total loading time: 0 Render date: 2024-07-10T10:36:18.102Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximal transference and summability of multilinear Fourier series

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  09 April 2009

Loukas Grafakos  and
Petr Honzík
Loukas Grafakos
Affiliation:
Department of Mathematics, University of Missouri, Columbia MO 65211, USA, e-mail: loukas@math.missouri.edu
Petr Honzík
Affiliation:
Department of Mathematics, Michigan State University, East Lansing MI 48824, USA, e-mail: petrhonz@math.msu.edu
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 obtain a maximal transference theorem that relates almost everywhere convergence of multilinear Fourier series to boundedness of maximal multilinear operators. We use this and other recent results on transference and multilinear operators to deduce Lp and almost everywhere summability of certain m–linear Fourier series. We formulate conditions for the convergence of multilinear series and we investigate certain kinds of summation.

MSC classification

Secondary: 42B15: Multipliers 42B25: Maximal functions, Littlewood-Paley theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 1 , February 2006 , pp. 65 - 80
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Blasco, O., ‘Bilinear multipliers and transference’, Int. J. Math. Math. Sci. 2005, 545–554.CrossRefGoogle Scholar
[2]Blasco, O. and Villaroya, F., ‘Transference of bilinear multiplier operators on Lorentz spaces’, Illinois J. Math. 47 (2003), 1327–1343.CrossRefGoogle Scholar
[3]Carleson, L., ‘On convergence and growth of partial sums of Fourier series’, Acta Math. 116 (1966), 135–157.CrossRefGoogle Scholar
[4]Coifman, R. R. and Weiss, G., Transference methods in analysis, Conference Board of the Mathematical Sciences Regional Conference Series in Math. 31 (Amer. Math. Soc., Providence, RI, 1976).Google Scholar
[5]de Leeuw, K., ‘On L p multipliers’, Ann. of Math. (2) 81 (1965), 364–379.CrossRefGoogle Scholar
[6]Fan, D. and Sato, S., ‘Transference of certain multilinear multiplier operators’, J. Austral. Math. Soc. 70 (2001), 37–55.CrossRefGoogle Scholar
[7]Grafakos, L., Classical and modern Fourier analysis (Prentice Hall, Upper Saddle River NJ, 2004).Google Scholar
[8]Grafakos, L. and Li, X., ‘The disc as a bilinear multiplier’, preprint.Google Scholar
[9]Grafakos, L. and Weiss, G., ‘Transference of multilinear operators’, Illinois J. Math. 40 (1996), 344–351.CrossRefGoogle Scholar
[10]Hunt, R., ‘On the convergence of Fourier series’, in: Orthogonal Expansions and Their Continuous Analogues (Edwardsville, IL, 1967) (ed. Haimo, D. T.) (Southern Illinois Univ. Press, Carbondale, IL, 1968) pp. 235–255.Google Scholar
[11]Kenig, C. and Tomas, P., ‘Maximal operators defined by Fourier multipliers’, Studia Math. 68 (1980), 79–83.CrossRefGoogle Scholar
[12]Lacey, M. and Thiele, C., ‘L p bounds for the bilinear Hilbert transform, p > 2’, Ann. of Math. (2) 146 (1997), 693–724.CrossRefGoogle Scholar
[13]Lacey, M. ‘On Calderón's conjecture’, Ann. of Math. (2) 149 (1999), 475–496.CrossRefGoogle Scholar
[14]Li, X. and Muscalu, C., ‘Generalizations of the Carleson-Hunt theorem. I. The classical singularity case’, preprint.Google Scholar
[15]Murray, M., ‘Multilinear convolutions and transference’, Michigan Math. J. 31 (1984), 321–330.CrossRefGoogle Scholar
[16]Muscalu, C., Thiele, C. and Tao, T., ‘A discrete model for the bi-Carleson operator’, Geom. Func. Anal. 12 (2002), 1324–1364.CrossRefGoogle Scholar