Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-16T17:50:32.714Z Has data issue: false hasContentIssue false
  • English
  • Français

Transference of Vector-valued Multipliers on Weighted Lp-spaces

Published online by Cambridge University Press:  20 November 2018

Oscar Blasco  and
Paco Villarroya
Oscar Blasco
Affiliation:
Centre for Mathematical Sciences, University of Lund, Lund, 22100, Sweden, e-mail: paco.villarroya@maths.lth.se
Paco Villarroya
Affiliation:
Centre for Mathematical Sciences, University of Lund, Lund, 22100, Sweden, e-mail: paco.villarroya@maths.lth.se
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.

New transference results for Fourier multiplier operators defined by regulated symbols are presented. We prove restriction and extension of multipliers between weighted Lebesgue spaces with two different weights, which belong to a class more general than periodic weights, and two different exponents of integrability that can be below one.

We also develop some ad-hoc methods that apply to weights defined by the product of periodic weights with functions of power type. Our vector-valued approach allows us to extend our results to transference of maximal multipliers and provide transference of Littlewood–Paley inequalities.

Keywords

42B1542B35Fourier multipliersrestriction theoremsweighted spaces
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 3 , 01 June 2013 , pp. 510 - 543
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Andersen, K. F. and Mohanty, P., Restriction and extension of Fourier multipliers between weighted Lp spaces on Rn and Tn. Proc. Amer. Math. Soc. 137(2009), no. 5, 1689–1697. http://dx.doi.org/10.1090/S0002-9939-08-09774-8 Google Scholar
[2] Berkson, E. and Gillespie, T. A., On restriction of multipliers in weighted settings. Indiana Univ. Math. J. 52(2003), no. 4, 927–961.Google Scholar
[3] Besicovitch, A. S., Analysis of conditions of almost periodicity. Acta Math. 58(1932), no. 1, 217–230. http://dx.doi.org/10.1007/BF02547778 Google Scholar
[4] Besicovitch, A. S. and Bohr, H., Almost periodicity and generalized trigonometric series. Acta Math. 57(1931), no. 1, 203–292. xhttp://dx.doi.org/10.1007/BF02403047 Google Scholar
[5] Besicovitch, A. S., On restriction of maximal multipliers on weighted settings. Trans. Amer. Math. Soc., to appear.Google Scholar
[6] Dabboucy, A. N. and Davis, H.W., A new characterization of Besicovitch almost periodic functions. Math. Scand. 28(1971), 341–354.Google Scholar
[7] Diestel, J., Jarchow, H., and Tonge, A., Absolutely summing operators. Cambrigde Studies in Advanced Mathematics, 43, Cambrigde University Press, Cambridge, 1995.Google Scholar
[8] García-Cuerva, J. and Rubiode Francia, J. L, Weighted norm inequalities and related topics. North–Holland Mathematical Studies, 116, Notas de Matemática, 104, North–Holland, Amsterdam, 1985.Google Scholar
[9] Jodeit, M. Jr., Restriction and extension of Fourier multipliers. Studia Math. 34(1970), 215–226.Google Scholar
[10] Kenig, C. E. and Tomas, P. A., Maximal operators defined by Fourier multipliers. Studia Math. 68(1980), no. 1, 79–83.Google Scholar
[11] Kurz, D. S., Littlewood-Paley and multipliers theorems on weighted Lp spaces. Trans. Amer. Math. Soc. 259(1980), no. 1, 235–254.Google Scholar
[12] de Leeuw, K., On Lp multipliers. Ann. of Math. (2) 81(1965), 364–379. http://dx.doi.org/10.2307/1970621 Google Scholar
[13] Raposo, J. A., Weak type (1; 1) multipliers on LCA groups. Studia Math. 122(1997), no. 2, 123–130.Google Scholar
[14] Stein, E. M. and Weiss, G., Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series, 32, Princeton University Press, Princeton, NJ, 1971.Google Scholar