# Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets

Annales de l'institut Fourier (1981)

- Volume: 31, Issue: 1, page 157-175
- ISSN: 0373-0956

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topSjögren, Peter, and Sjölin, Per. "Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets." Annales de l'institut Fourier 31.1 (1981): 157-175. <http://eudml.org/doc/74479>.

@article{Sjögren1981,

abstract = {Let $E\subset \{\bf R\}$ be a closed null set. We prove an equivalence between the Littlewood-Paley decomposition in $L^p$ with respect to the complementary intervals of $E$ and Fourier multipliers of Hörmander-Mihlin and Marcinkiewicz type with singularities on $E$. Similar properties are studied in $\{\bf R\}^2$ for a union of rays from the origin. Then there are connections with the maximal function operator with respect to all rectangles parallel to these rays. In particular, this maximal operator is proved to be bounded on $L^p$, $1< p< \infty $, when the rays form an iterated lacunary sequence.},

author = {Sjögren, Peter, Sjölin, Per},

journal = {Annales de l'institut Fourier},

keywords = {Lp spaces; Fourier multipliers; Littlewood-Paley decompositions; maximal functions},

language = {eng},

number = {1},

pages = {157-175},

publisher = {Association des Annales de l'Institut Fourier},

title = {Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets},

url = {http://eudml.org/doc/74479},

volume = {31},

year = {1981},

}

TY - JOUR

AU - Sjögren, Peter

AU - Sjölin, Per

TI - Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets

JO - Annales de l'institut Fourier

PY - 1981

PB - Association des Annales de l'Institut Fourier

VL - 31

IS - 1

SP - 157

EP - 175

AB - Let $E\subset {\bf R}$ be a closed null set. We prove an equivalence between the Littlewood-Paley decomposition in $L^p$ with respect to the complementary intervals of $E$ and Fourier multipliers of Hörmander-Mihlin and Marcinkiewicz type with singularities on $E$. Similar properties are studied in ${\bf R}^2$ for a union of rays from the origin. Then there are connections with the maximal function operator with respect to all rectangles parallel to these rays. In particular, this maximal operator is proved to be bounded on $L^p$, $1< p< \infty $, when the rays form an iterated lacunary sequence.

LA - eng

KW - Lp spaces; Fourier multipliers; Littlewood-Paley decompositions; maximal functions

UR - http://eudml.org/doc/74479

ER -

## References

top- [1] A. CORDOBA and C. FEFFERMAN, A weighted norm inequality for singular integrals, Studia Math., 57 (1976), 97-101. Zbl0356.44003MR54 #8132
- [2] A. CORDOBA and R. FEFFERMAN, On the equivalence between the boundedness of certain classes of maximal and multiplier operators in Fourier analysis, Proc. Natl. Acad. Sci. USA, 74 (1977), 423-425. Zbl0342.42003MR55 #6096
- [3] M. JODEIT Jr, A note on Fourier multipliers, Proc. Amer. Math. Soc., 27 (1971), 423-424. Zbl0214.13301MR42 #4965
- [4] D.S. KURTZ and R.L. WHEEDEN, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc., 255 (1979), 343-362. Zbl0427.42004MR81j:42021
- [5] A. NAGEL, E.M. STEIN and S. WAINGER, Differentiation in lacunary directions, Proc. Natl. Acad. Sci. USA, 75 (1978), 1060-1062. Zbl0391.42015MR57 #6349
- [6] J.L. RUBIO de FRANCIA, Vector valued inequalities for operators in Lp spaces, Bull. London Math. Soc., 12 (1980), 211-215. Zbl0417.47010MR81g:42024
- [7] E.M. STEIN, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, 1970. Zbl0207.13501MR44 #7280

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