Multiple singular integrals and maximal functions along hypersurfaces

Javier Duoandikoetxea

Annales de l'institut Fourier (1986)

  • Volume: 36, Issue: 4, page 185-206
  • ISSN: 0373-0956

Abstract

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Maximal functions written as convolution with a multiparametric family of positive measures, and singular integrals whose kernel is decomposed as a multiple series of measures, are shown to be bounded in L p , 1 < p < ∞ . The proofs are based on the decomposition of the operators according to the size of the Fourier transform of the measures, assuming some regularity at zero and decay at infinity of these Fourier transforms. Applications are given to homogeneous singular integrals in product spaces with size conditions on the kernel and maximal functions and multiple Hilbert transforms along different types of surfaces.

How to cite

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Duoandikoetxea, Javier. "Multiple singular integrals and maximal functions along hypersurfaces." Annales de l'institut Fourier 36.4 (1986): 185-206. <http://eudml.org/doc/74734>.

@article{Duoandikoetxea1986,
abstract = {Maximal functions written as convolution with a multiparametric family of positive measures, and singular integrals whose kernel is decomposed as a multiple series of measures, are shown to be bounded in $L^ p$, $1&lt; p&lt; \infty $. The proofs are based on the decomposition of the operators according to the size of the Fourier transform of the measures, assuming some regularity at zero and decay at infinity of these Fourier transforms. Applications are given to homogeneous singular integrals in product spaces with size conditions on the kernel and maximal functions and multiple Hilbert transforms along different types of surfaces.},
author = {Duoandikoetxea, Javier},
journal = {Annales de l'institut Fourier},
keywords = {Maximal functions; singular integrals; Hilbert transforms; surfaces},
language = {eng},
number = {4},
pages = {185-206},
publisher = {Association des Annales de l'Institut Fourier},
title = {Multiple singular integrals and maximal functions along hypersurfaces},
url = {http://eudml.org/doc/74734},
volume = {36},
year = {1986},
}

TY - JOUR
AU - Duoandikoetxea, Javier
TI - Multiple singular integrals and maximal functions along hypersurfaces
JO - Annales de l'institut Fourier
PY - 1986
PB - Association des Annales de l'Institut Fourier
VL - 36
IS - 4
SP - 185
EP - 206
AB - Maximal functions written as convolution with a multiparametric family of positive measures, and singular integrals whose kernel is decomposed as a multiple series of measures, are shown to be bounded in $L^ p$, $1&lt; p&lt; \infty $. The proofs are based on the decomposition of the operators according to the size of the Fourier transform of the measures, assuming some regularity at zero and decay at infinity of these Fourier transforms. Applications are given to homogeneous singular integrals in product spaces with size conditions on the kernel and maximal functions and multiple Hilbert transforms along different types of surfaces.
LA - eng
KW - Maximal functions; singular integrals; Hilbert transforms; surfaces
UR - http://eudml.org/doc/74734
ER -

References

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  1. [1] A. P. CALDERON, A. ZYGMUND, On singular integrals, Amer. J. Math., 78 (1956), 289-309. Zbl0072.11501MR18,894a
  2. [2] H. CARLSSON, P. SJÖGREN, Estimates for maximal functions along hypersurfaces, Univ. of Göteborg, preprint, 1984. Zbl0629.42010
  3. [3] H. CARLSSON, P. SJÖGREN, J. O. STROMBERG, Multiparameter maximal functions along dilation invariant hypersurfaces, Univ. of Göteborg, preprint, 1984. Zbl0578.42018
  4. [4] J. DUOANDIKOETXEA, J. L. RUBIO DE FRANCIA, Maximal and singular integral operators via Fourier transform estimates, Invent. Math., 84 (1986), 541-561. Zbl0568.42012MR87f:42046
  5. [5] R. FEFFERMAN, Singular integrals on product domains, Bull. Amer. Math. Soc., 4 (1981), 195-201. Zbl0466.42007MR83i:42014
  6. [6] R. FEFFERMAN, E. M. STEIN, Singular integrals on product spaces, Adv. in Math., 45 (1982), 117-143. Zbl0517.42024MR84d:42023
  7. [7] D. S. KURTZ, Littlewood-Paley and multiplier theorems on weighted Lp spaces, Trans. Amer. Math. Soc., 259 (1980), 235-254. Zbl0436.42012MR80f:42013
  8. [8] A. NAGEL, S. WAINGER, L2-boundedness of Hilbert transforms along surfaces and convolution operators homogeneous with respect to a multiple parameter group, Amer. J. Math., 99 (1977), 761-785. Zbl0374.44003MR56 #9192
  9. [9] J. L. RUBIO DE FRANCIA, Factorization theory and Ap weights, Amer. J. Math., 106 (1984), 533-547. Zbl0558.42012MR86a:47028a
  10. [10] E. M. STEIN, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton N.J., 1970. Zbl0207.13501MR44 #7280
  11. [11] E. M. STEIN, S. WAINGER, Problems in Harmonic Analysis related to curvature, Bull. Amer. Math. Soc., 84 (1978), 1239-1295. Zbl0393.42010MR80k:42023
  12. [12] R. S. STRICHARTZ, Singular integrals supported in manifolds, Studia Math., 74 (1982), 137-151. Zbl0501.43007MR85c:42019
  13. [13] J. T. VANCE, Lp-boundedness of the multiple Hilbert transform along a surface, Pacific J. Math., 108 (1983), 221-241. Zbl0462.44001MR85h:44010

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