Two problems of Calderón-Zygmund theory on product-spaces

Jean-Lin Journé

Annales de l'institut Fourier (1988)

  • Volume: 38, Issue: 1, page 111-132
  • ISSN: 0373-0956

Abstract

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R. Fefferman has shown that, on a product-space with two factors, an operator T bounded on L 2 maps L ∞ into BMO of the product if the mean oscillation on a rectangle R of the image of a bounded function supported out of a multiple R’ of R, is dominated by C | R | s | R | - s , for some s > 0 . We show that this result does not extend in general to the case where E has three or more factors but remains true in this case if in addition T is a convolution operator, provided s > s 0 ( E ) . We also show that the Calderon-Coifman bicommutators, obtained from the Calderon commutators by multilinear tensor product, are bounded on L 2 with norms growing polynomially.

How to cite

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Journé, Jean-Lin. "Two problems of Calderón-Zygmund theory on product-spaces." Annales de l'institut Fourier 38.1 (1988): 111-132. <http://eudml.org/doc/74785>.

@article{Journé1988,
abstract = {R. Fefferman has shown that, on a product-space with two factors, an operator T bounded on $L^ 2$ maps $L^\{\infty \}$ into BMO of the product if the mean oscillation on a rectangle R of the image of a bounded function supported out of a multiple R’ of R, is dominated by $C\vert R\vert ^ s\vert R\vert ^\{-s\}$, for some $s&gt;0$. We show that this result does not extend in general to the case where E has three or more factors but remains true in this case if in addition T is a convolution operator, provided $s&gt;s_ 0(E)$. We also show that the Calderon-Coifman bicommutators, obtained from the Calderon commutators by multilinear tensor product, are bounded on $L^ 2$ with norms growing polynomially.},
author = {Journé, Jean-Lin},
journal = {Annales de l'institut Fourier},
keywords = {mean oscillation; convolution operator; Calderon-Coifman bicommutators; multilinear tensor product},
language = {eng},
number = {1},
pages = {111-132},
publisher = {Association des Annales de l'Institut Fourier},
title = {Two problems of Calderón-Zygmund theory on product-spaces},
url = {http://eudml.org/doc/74785},
volume = {38},
year = {1988},
}

TY - JOUR
AU - Journé, Jean-Lin
TI - Two problems of Calderón-Zygmund theory on product-spaces
JO - Annales de l'institut Fourier
PY - 1988
PB - Association des Annales de l'Institut Fourier
VL - 38
IS - 1
SP - 111
EP - 132
AB - R. Fefferman has shown that, on a product-space with two factors, an operator T bounded on $L^ 2$ maps $L^{\infty }$ into BMO of the product if the mean oscillation on a rectangle R of the image of a bounded function supported out of a multiple R’ of R, is dominated by $C\vert R\vert ^ s\vert R\vert ^{-s}$, for some $s&gt;0$. We show that this result does not extend in general to the case where E has three or more factors but remains true in this case if in addition T is a convolution operator, provided $s&gt;s_ 0(E)$. We also show that the Calderon-Coifman bicommutators, obtained from the Calderon commutators by multilinear tensor product, are bounded on $L^ 2$ with norms growing polynomially.
LA - eng
KW - mean oscillation; convolution operator; Calderon-Coifman bicommutators; multilinear tensor product
UR - http://eudml.org/doc/74785
ER -

References

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  1. [1] R. FEFFERMAN, Calderón-Zygmund theory for product domains-Hp spaces, P.N.S.A., vol. 83 (1986), 840-843. Zbl0602.42023MR87h:42032
  2. [2] J. PIPHER, Journé's covering lemma and its extension to higher dimensions, Duke Journal of Math, 53 n° 3 (1986), 683-690. Zbl0645.42018MR88a:42019
  3. [3] J.-L. JOURNE, Calderón-Zygmund operators on product spaces, Mat. Iberoamerica, vol. 1, n° 3 (1985), 55-91. Zbl0634.42015MR88d:42028
  4. [4] R. FEFFERMAN, Harmonic analysis on product spaces, Ann. of Math., (2), vol. 126 (1987), 109-130. Zbl0644.42017MR90e:42030
  5. [5] R. FEFFERMAN, Functions of bounded mean oscillation on the polydisc, Ann. of Math., (2), 10 (1979), 395-406. Zbl0429.32016MR81c:32016
  6. [6] L. CARLESON, A counterexample for measures bounded for Hp for the bi-disc, Mittag Leffer Report n° 7, 1974. 
  7. [7] S.-Y. A. CHANG and R. FEFFERMAN, A continuous version of the duality of H1 and BMO on the bi-disc, Ann. of Math., (2), 112 (1980), 179-201. Zbl0451.42014MR82a:32009
  8. [8] J.-L. RUBIO DE FRANCIA, A Littlewood-Paley inequality for arbitrary intervals, Revista Mat. Iberoamericana, vol. 1, n° 2 (1985). Zbl0611.42005MR87j:42057
  9. [9] P. KRIKELES, Ph. D. dissertation, Yale University, 1982. 
  10. [10] M. CHRIST, and J.-L. JOURNÉ, Polynomial growth estimates for multilinear singular integral operators, Acta Math., 159 (1987), 51-80. Zbl0645.42017MR89a:42024
  11. [11] S.-Y. A. CHANG, Carleson measure on the bi-disc, Ann. of Math., (2), 109 (1979), 613-620. Zbl0401.28004MR80j:32009
  12. [12] J.-L. JOURNÉ, Calderón-Zygmund operators, pseudo-differential operators and the Cauchy integral of Calderón, L.N. 994, Springer-Verlag. Zbl0508.42021MR85i:42021
  13. [13] S.-Y. A. CHANG and R. FEFFERMAN, The Calderón-Zygmund decomposition on product domains, Am. J. of Math., vol. 104, 3, 455-468. Zbl0513.42019MR84a:42028

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