Necessary condition for measures which are ( L q , L p ) multipliers

Bérenger Akon Kpata[1]; Ibrahim Fofana[1]; Konin Koua[1]

  • [1] UFR Mathématiques et Informatique Université de Cocody 22 BP 582 Abidjan 22 Côte d’Ivoire

Annales mathématiques Blaise Pascal (2009)

  • Volume: 16, Issue: 2, page 339-353
  • ISSN: 1259-1734

Abstract

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Let G be a locally compact group and ρ the left Haar measure on G . Given a non-negative Radon measure μ , we establish a necessary condition on the pairs q , p for which μ is a multiplier from L q G , ρ to L p G , ρ . Applied to ℝ n , our result is stronger than the necessary condition established by Oberlin in [14] and is closely related to a class of measures defined by Fofana in [7].When G is the circle group, we obtain a generalization of a condition stated by Oberlin [15] and improve on it in some cases.

How to cite

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Kpata, Bérenger Akon, Fofana, Ibrahim, and Koua, Konin. "Necessary condition for measures which are $(L^{q},L^{p})$ multipliers." Annales mathématiques Blaise Pascal 16.2 (2009): 339-353. <http://eudml.org/doc/10584>.

@article{Kpata2009,
abstract = {Let $G$ be a locally compact group and $\rho $ the left Haar measure on $G$. Given a non-negative Radon measure $\mu $, we establish a necessary condition on the pairs $\left( q,\ p\right) $ for which $\mu $ is a multiplier from $L^\{q\}\left( G,\ \rho \right) $ to $L^\{p\}\left( G,\ \rho \right) $. Applied to $\mathbb\{R\}^\{n\}$, our result is stronger than the necessary condition established by Oberlin in [14] and is closely related to a class of measures defined by Fofana in [7].When $G$ is the circle group, we obtain a generalization of a condition stated by Oberlin [15] and improve on it in some cases.},
affiliation = {UFR Mathématiques et Informatique Université de Cocody 22 BP 582 Abidjan 22 Côte d’Ivoire; UFR Mathématiques et Informatique Université de Cocody 22 BP 582 Abidjan 22 Côte d’Ivoire; UFR Mathématiques et Informatique Université de Cocody 22 BP 582 Abidjan 22 Côte d’Ivoire},
author = {Kpata, Bérenger Akon, Fofana, Ibrahim, Koua, Konin},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Cantor-Lebesgue measure; $L^\{q\}$-improving measure; non-negative Radon measure; Cantor–Lebesgue measure; Radon measure; multiplier},
language = {eng},
month = {7},
number = {2},
pages = {339-353},
publisher = {Annales mathématiques Blaise Pascal},
title = {Necessary condition for measures which are $(L^\{q\},L^\{p\})$ multipliers},
url = {http://eudml.org/doc/10584},
volume = {16},
year = {2009},
}

TY - JOUR
AU - Kpata, Bérenger Akon
AU - Fofana, Ibrahim
AU - Koua, Konin
TI - Necessary condition for measures which are $(L^{q},L^{p})$ multipliers
JO - Annales mathématiques Blaise Pascal
DA - 2009/7//
PB - Annales mathématiques Blaise Pascal
VL - 16
IS - 2
SP - 339
EP - 353
AB - Let $G$ be a locally compact group and $\rho $ the left Haar measure on $G$. Given a non-negative Radon measure $\mu $, we establish a necessary condition on the pairs $\left( q,\ p\right) $ for which $\mu $ is a multiplier from $L^{q}\left( G,\ \rho \right) $ to $L^{p}\left( G,\ \rho \right) $. Applied to $\mathbb{R}^{n}$, our result is stronger than the necessary condition established by Oberlin in [14] and is closely related to a class of measures defined by Fofana in [7].When $G$ is the circle group, we obtain a generalization of a condition stated by Oberlin [15] and improve on it in some cases.
LA - eng
KW - Cantor-Lebesgue measure; $L^{q}$-improving measure; non-negative Radon measure; Cantor–Lebesgue measure; Radon measure; multiplier
UR - http://eudml.org/doc/10584
ER -

References

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  1. William Beckner, Svante Janson, David Jerison, Convolution inequalities on the circle, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) (1983), 32-43, Wadsworth, Belmont, CA MR730056
  2. A. Bonami, Étude des coefficients de Fourier des fonctions de L p G , Ann. Inst. Fourier (Grenoble) 20 (1970), 335-402 Zbl0195.42501MR283496
  3. M. Christ, A convolution inequality concerning Cantor-Lebesgue measures, Revista Mat. Iberoamericana vol. 1, n. ∘ 4 (1985), 79-83 Zbl0644.42011MR850410
  4. K. J Falconer, The geometry of fractal sets, (1985), Cambridge University Press, London/New York Zbl0587.28004MR867284
  5. K. J Falconer, Fractal geometry, (1990), Wiley, New York Zbl0689.28003MR1102677
  6. I. Fofana, Continuité de l’intégrale fractionnaire et espaces L q , l p α , C. R. A. S. Paris t. 308, série I (1989), 525-527 Zbl0669.42004MR1001045
  7. I. Fofana, Transformation de Fourier dans L q , l p α et M p , α , Afrika matematika série 3, vol. 5 (1995), 53-76 Zbl0885.42005MR1431325
  8. I. Fofana, Espaces L q , l p α et Continuité de l’opérateur maximal fractionnaire de Hardy-Littlewood, Afrika matematika série 3, vol. 12 (2001), 23-37 Zbl1026.42020MR1876792
  9. C. C. Graham, K. Hare, D. Ritter, The size of L p -improving measures, J. Funct. Anal. 84 (1989), 472-495 Zbl0678.43001MR1001469
  10. R. Larsen, An introduction to the theory of multipliers, (1971), Springer-Verlag, Berlin, Heidelberg, New York Zbl0213.13301MR435738
  11. K-S. Lau, Fractal Measures and Mean ρ -Variations, J. Funct. Anal. 108 (1992), 427-457 Zbl0767.28007MR1176682
  12. D. M. Oberlin, A convolution property of the Cantor-Lebesgue measure, Colloq. Math. 47 (1982), 113-117 Zbl0501.42007MR679392
  13. D. M. Oberlin, Convolution with measure on hypersurfaces, Math. Proc. Camb. Phil. Soc. 129 (2000), 517-526 Zbl0972.42009MR1780502
  14. D. M. Oberlin, Affine dimension : measuring the vestiges of curvature, Michigan Math. J. 51 (2003), 13-26 Zbl1035.53023MR1960918
  15. D. M. Oberlin, A convolution property of the Cantor-Lebesgue measure II, Colloq. Math. 97 (2003), 23-28 Zbl1095.42007MR2010539
  16. D. Ritter, Most Riesz product measures are L p -improving, Proc. Amer. Math. Soc. 97 (1986), 291-295 Zbl0593.43002MR835883
  17. D. Ritter, Some singular measures on the circle which improve L p spaces, Colloq. Math. 52 (1987), 133-144 Zbl0637.43002MR891505
  18. E. M. Stein, Harmonic Analysis on ℝ n , 13 (1976), 97-135, Studies in Harmonic Analysis, MAA Studies in Mathematics Zbl0337.42016MR461002
  19. A. Zygmund, Trigonometric series. 2nd ed. Vol. I, (1959), Cambridge University Press, New York Zbl0085.05601MR107776

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