On certain nonstandard Calderón-Zygmund operators

Steve Hofmann

Studia Mathematica (1994)

  • Volume: 109, Issue: 2, page 105-131
  • ISSN: 0039-3223

Abstract

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We formulate a version of the T1 theorem which enables us to treat singular integrals whose kernels need not satisfy the usual smoothness conditions. We also prove a weighted version. As an application of the general theory, we consider a class of multilinear singular integrals in ℝ n related to the first Calderón commutator, but with a kernel which is far less regular.

How to cite

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Hofmann, Steve. "On certain nonstandard Calderón-Zygmund operators." Studia Mathematica 109.2 (1994): 105-131. <http://eudml.org/doc/216064>.

@article{Hofmann1994,
abstract = {We formulate a version of the T1 theorem which enables us to treat singular integrals whose kernels need not satisfy the usual smoothness conditions. We also prove a weighted version. As an application of the general theory, we consider a class of multilinear singular integrals in $ℝ^n$ related to the first Calderón commutator, but with a kernel which is far less regular.},
author = {Hofmann, Steve},
journal = {Studia Mathematica},
keywords = {nonstandard Calderón-Zygmund operators; Muckenhoupt class ; variant of the theorem; Calderón commutator; rough operators},
language = {eng},
number = {2},
pages = {105-131},
title = {On certain nonstandard Calderón-Zygmund operators},
url = {http://eudml.org/doc/216064},
volume = {109},
year = {1994},
}

TY - JOUR
AU - Hofmann, Steve
TI - On certain nonstandard Calderón-Zygmund operators
JO - Studia Mathematica
PY - 1994
VL - 109
IS - 2
SP - 105
EP - 131
AB - We formulate a version of the T1 theorem which enables us to treat singular integrals whose kernels need not satisfy the usual smoothness conditions. We also prove a weighted version. As an application of the general theory, we consider a class of multilinear singular integrals in $ℝ^n$ related to the first Calderón commutator, but with a kernel which is far less regular.
LA - eng
KW - nonstandard Calderón-Zygmund operators; Muckenhoupt class ; variant of the theorem; Calderón commutator; rough operators
UR - http://eudml.org/doc/216064
ER -

References

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  1. [BC] B. Bajsanski and R. Coifman, On singular integrals, in: Proc. Sympos. Pure Math. 10, Amer. Math. Soc., Providence, 1967, 1-17. 
  2. [Ca] A. P. Calderón, Commutators of singular integrals, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092-1099. Zbl0151.16901
  3. [CZ] A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289-309. Zbl0072.11501
  4. [CC] C. P. Calderón, On commutators of singular integrals, Studia Math. 53 (1975), 139-174. Zbl0315.44006
  5. [CS] A. Carbery and A. Seeger, Conditionally convergent series of linear operators on L p -spaces and L p -estimates for pseudodifferential operators, Proc. London Math. Soc. 57 (1988), 481-510. Zbl0681.35091
  6. [C] M. Christ, Weak type (1,1) bounds for rough operators, Ann. of Math. 128 (1988), 19-42. Zbl0666.47027
  7. [C2] M. Christ, Hilbert transforms along curves I. Nilpotent groups, ibid. 122 (1985), 575-596. Zbl0593.43011
  8. [Co] J. Cohen, A sharp estimate for a multilinear singular integral in ℝ n , Indiana Univ. Math. J. 30 (1981), 693-702. Zbl0596.42004
  9. [CDMS] R. Coifman, G. David, Y. Meyer and S. Semmes, ω-Calderón-Zygmund operators, in: Proc. Conf. Harmonic Analysis and PDE, El Escorial 1987, Lecture Notes in Math. 1384, Springer, Berlin, 1989, 132-145. 
  10. [CRW] R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635. Zbl0326.32011
  11. [DJ] G. David and J.-L. Journé, A boundedness criterion for generalized Calderón-Zygmund operators, ibid. 120 (1984), 371-397. Zbl0567.47025
  12. [DJS] G. David, J.-L. Journé and S. Semmes, Calderón-Zygmund operators, para-accretive functions and interpolation, preprint. Zbl0604.42014
  13. [DR] J. Duoandikoetxea and J. L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), 541-561. Zbl0568.42012
  14. [GR] J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud. 116, North-Holland, Amsterdam, 1985. 
  15. [HH] Y. S. Han and S. Hofmann, T1 theorems for Besov and Triebel-Lizorkin spaces, Trans. Amer. Math. Soc., to appear. Zbl0779.42010
  16. [HS] Y. S. Han and E. T. Sawyer, Para-accretive functions, the weak boundedness property, and interpolation, Rev. Mat. Iberoamericana, to appear. 
  17. [H] S. Hofmann, Weighted inequalities for commutators of rough singular integrals, Indiana Univ. Math. J. 39 (1990), 1275-1304. Zbl0708.42012
  18. [H2] S. Hofmann, Singular integrals of Calderón-type in ℝ n , and BMO, Rev. Mat. Iberoamericana, to appear. 
  19. [Hu] Y. Hu, An estimate for multilinear singular integrals on ℝ n , Beijing Daxue Xuebao 1985 (3), 19-26 (in Chinese, with English summary reprinted in MR 87h:42026). 
  20. [J] J.-L. Journé, Calderón-Zygmund Operators, Pseudo-Differential Operators, and the Cauchy Integral of Calderón, Lecture Notes in Math. 994, Springer, Berlin, 1983. Zbl0508.42021
  21. [M] Y. Meyer, La continuité des opérateurs définis par des intégrales singulières, Monografías de Matemáticas V. 4, Univ. Autónoma de Madrid. Zbl0547.47032
  22. [S] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970. Zbl0207.13501

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