Singular integral characterization of nonisotropic generalized BMO spaces

Raquel Crescimbeni

Commentationes Mathematicae Universitatis Carolinae (2007)

  • Volume: 48, Issue: 2, page 225-238
  • ISSN: 0010-2628

Abstract

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We extend a result of Coifman and Dahlberg [Singular integral characterizations of nonisotropic H p spaces and the F. and M. Riesz theorem, Proc. Sympos. Pure Math., Vol. 35, pp. 231–234; Amer. Math. Soc., Providence, 1979] on the characterization of H p spaces by singular integrals of ℝ n with a nonisotropic metric. Then we apply it to produce singular integral versions of generalized BMO spaces. More precisely, if T λ is the family of dilations in ℝ n induced by a matrix with a nonnegative eigenvalue, then there exist 2 n singular integral operators homogeneous with respect to the dilations T λ that characterize BMO ϕ under a natural condition on ϕ .

How to cite

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Crescimbeni, Raquel. "Singular integral characterization of nonisotropic generalized BMO spaces." Commentationes Mathematicae Universitatis Carolinae 48.2 (2007): 225-238. <http://eudml.org/doc/250206>.

@article{Crescimbeni2007,
abstract = {We extend a result of Coifman and Dahlberg [Singular integral characterizations of nonisotropic $H^p$ spaces and the F. and M. Riesz theorem, Proc. Sympos. Pure Math., Vol. 35, pp. 231–234; Amer. Math. Soc., Providence, 1979] on the characterization of $H^p$ spaces by singular integrals of $\mathbb \{R\}^n$ with a nonisotropic metric. Then we apply it to produce singular integral versions of generalized BMO spaces. More precisely, if $T_\lambda $ is the family of dilations in $\mathbb \{R\}^n$ induced by a matrix with a nonnegative eigenvalue, then there exist $2n$ singular integral operators homogeneous with respect to the dilations $T_\lambda $ that characterize BMO$_\varphi $ under a natural condition on $\varphi $.},
author = {Crescimbeni, Raquel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {singular integral; nonisotropic generalized BMO; singular integral; nonisotropic generalized BMO},
language = {eng},
number = {2},
pages = {225-238},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Singular integral characterization of nonisotropic generalized BMO spaces},
url = {http://eudml.org/doc/250206},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Crescimbeni, Raquel
TI - Singular integral characterization of nonisotropic generalized BMO spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 2
SP - 225
EP - 238
AB - We extend a result of Coifman and Dahlberg [Singular integral characterizations of nonisotropic $H^p$ spaces and the F. and M. Riesz theorem, Proc. Sympos. Pure Math., Vol. 35, pp. 231–234; Amer. Math. Soc., Providence, 1979] on the characterization of $H^p$ spaces by singular integrals of $\mathbb {R}^n$ with a nonisotropic metric. Then we apply it to produce singular integral versions of generalized BMO spaces. More precisely, if $T_\lambda $ is the family of dilations in $\mathbb {R}^n$ induced by a matrix with a nonnegative eigenvalue, then there exist $2n$ singular integral operators homogeneous with respect to the dilations $T_\lambda $ that characterize BMO$_\varphi $ under a natural condition on $\varphi $.
LA - eng
KW - singular integral; nonisotropic generalized BMO; singular integral; nonisotropic generalized BMO
UR - http://eudml.org/doc/250206
ER -

References

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  1. Coifman R., Dahlberg B., Singular integral characterizations of nonisotropic H p spaces and the F. and M. Riesz theorem, Proc. Sympos. Pure Math., Vol. 35, pp.231-234; Amer. Math. Soc., Providence, 1979. MR0545260
  2. Coifman R., Weiss G., Analyse harmonique non-conmutative sur certain espaces homogenes, Lecture Notes in Mathematics 242, Springer, Berlin-New York, 1971. MR0499948
  3. Crescimbeni R., Singular integral characterization of functions with conditions on the mean oscillation on spaces of homogeneous type, Rev. Un. Mat. Argentina 39 153-171 (1995). (1995) Zbl0892.42007MR1376792
  4. Fefferman C., Stein E., H p spaces of several variables, Acta Math. 129 137-193 (1972). (1972) MR0447953
  5. de Guzmán M., Real Variable Methods in Fourier Analysis, Mathematics Studies 46, North Holland, Amsterdam, 1981. MR0596037
  6. Krasnosel'skii M., Rutickii J., Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961. MR0126722
  7. Macías R., Segovia C., Lipschitz functions on spaces of homogeneous type, Adv. in Math. 33 257-270 (1979). (1979) MR0546295
  8. Macías R., Segovia C., A decomposition into atoms of distributions on spaces of homogeneous type, Adv. in Math. 33 271-309 (1979). (1979) MR0546296
  9. Viviani B., An atomic decomposition of the predual of B M O ( ρ ) , Rev. Mat. Iberoamericana 3 3-4 401-425 (1987). (1987) Zbl0665.46022MR0996824

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