Advertisement

Arkiv för Matematik

, Volume 54, Issue 1, pp 157–180 | Cite as

Weighted weak type (1,1) estimates for singular integrals with non-isotropic homogeneity

  • Shuichi SatoEmail author
Article
  • 134 Downloads

Abstract

We prove sharp weighted weak type (1,1) estimates for rough singular integral operators on homogeneous groups. Similar results are shown for singular integrals on \(\mathbb{R}^{2}\) with the generalized homogeneity.

Keywords

Singular Integral Singular Integral Operator Weak Type Weight Norm Inequality Generalize Homogeneity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Calderón, A. P., Inequalities for the maximal function relative to a metric, Studia Math. 57 (1976), 297–306. MathSciNetzbMATHGoogle Scholar
  2. 2.
    Calderón, A. P. and Torchinsky, A., Parabolic maximal functions associated with a distribution, Adv. Math. 16 (1975), 1–64. MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Calderón, A. P. and Zygmund, A., On singular integrals, Amer. J. Math. 78 (1956), 289–309. MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Carbery, A., Hernández, E. and Soria, F., Estimates for the Kakeya maximal operator on radial functions in \(\mathbb{R}^{n}\), in Harmonic Analysis, ICM-90 Satellite Conference Proceedings, pp. 41–50, Springer, Tokyo, 1991. CrossRefGoogle Scholar
  5. 5.
    Christ, M., Hilbert transforms along curves I. Nilpotent groups, Ann. Math. 122 (1985), 575–596. MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Christ, M., Weak type (1,1) bounds for rough operators, Ann. Math. 128 (1988), 19–42. MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Christ, M. and Rubio de Francia, J. L., Weak type (1,1) bounds for rough operators, II, Invent. Math. 93 (1988), 225–237. MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Coifman, R. R. and Weiss, G., Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes, Lecture Notes in Math. 242, Springer, Berlin and New York, 1971. zbMATHGoogle Scholar
  9. 9.
    Duoandikoetxea, J., Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc. 336 (1993), 869–880. MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Duoandikoetxea, J. and Rubio de Francia, J. L., Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), 541–561. MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fan, D. and Sato, S., Weak type (1,1) estimates for Marcinkiewicz integrals with rough kernels, Tohoku Math. J. 53 (2001), 265–284. MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fan, D. and Sato, S., Weighted weak type (1,1) estimates for singular integrals and Littlewood-Paley functions, Studia Math. 163 (2004), 119–136. MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Folland, G. B. and Stein, E. M., Hardy Spaces on Homogeneous Groups, Princeton Univ. Press, Princeton, NJ, 1982. zbMATHGoogle Scholar
  14. 14.
    Garcia-Cuerva, J. and Rubio de Francia, J. L., Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985. zbMATHGoogle Scholar
  15. 15.
    Hofmann, S., Weak (1,1) boundedness of singular integrals with nonsmooth kernel, Proc. Amer. Math. Soc. 103 (1988), 260–264. MathSciNetzbMATHGoogle Scholar
  16. 16.
    Hofmann, S., Weighted weak-type (1,1) inequalities for rough operators, Proc. Amer. Math. Soc. 107 (1989), 423–435. MathSciNetzbMATHGoogle Scholar
  17. 17.
    Hofmann, S., Weighted norm inequalities and vector-valued inequalities for certain rough operators, Indiana Univ. Math. J. 42 (1993), 1–14. MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Koranyi, A. and Vagi, S., Singular integrals on homogeneous spaces and some problems of classical analysis, Ann. Sc. Norm. Super. Pisa 25 (1971), 575–648. MathSciNetzbMATHGoogle Scholar
  19. 19.
    Muckenhoupt, B. and Wheeden, R. L., Weighted norm inequalities for singular and fractional integrals, Trans. Amer. Math. Soc. 161 (1971), 249–258. MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nagel, A. and Stein, E. M., Lectures on Pseudo-Differential Operators, Mathematical Notes 24, Princeton University Press, Princeton, NJ, 1979. zbMATHGoogle Scholar
  21. 21.
    Rivière, N., Singular integrals and multiplier operators, Ark. Mat. 9 (1971), 243–278. MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sato, S., Estimates for singular integrals along surfaces of revolution, J. Aust. Math. Soc. 86 (2009), 413–430. MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sato, S., A note on L p estimates for singular integrals, Sci. Math. Jpn. 71 (2010), 343–348. MathSciNetzbMATHGoogle Scholar
  24. 24.
    Sato, S., Weak type (1,1) estimates for parabolic singular integrals, Proc. Edinb. Math. Soc. 54 (2011), 221–247. MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sato, S., Estimates for singular integrals on homogeneous groups, J. Math. Anal. Appl. 400 (2013), 311–330. MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Seeger, A., Singular integral operators with rough convolution kernels, J. Amer. Math. Soc. 9 (1996), 95–105. MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Soria, F. and Weiss, G., A remark on singular integrals and power weights, Indiana Univ. Math. J. 43 (1994), 187–204. MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Stein, E. M. and Wainger, S., Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. (N.S.) 84 (1978), 1239–1295. MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tao, T., The weak-type (1,1) of LlogL homogeneous convolution operator, Indiana Univ. Math. J. 48 (1999), 1547–1584. MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Vargas, A., Weighted weak type (1,1) bounds for rough operators, J. Lond. Math. Soc. (2) 54 (1996), 297–310. MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Watson, D., Weighted estimates for singular integrals via Fourier transform estimates, Duke Math. J. 60 (1990), 389–399. MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institut Mittag-Leffler 2015

Authors and Affiliations

  1. 1.Department of Mathematics Faculty of EducationKanazawa UniversityKanazawaJapan

Personalised recommendations