# Geometric Fourier analysis

Annales de l'institut Fourier (1982)

- Volume: 32, Issue: 3, page 215-226
- ISSN: 0373-0956

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topCordoba, Antonio. "Geometric Fourier analysis." Annales de l'institut Fourier 32.3 (1982): 215-226. <http://eudml.org/doc/74547>.

@article{Cordoba1982,

abstract = {In this paper we continue the study of the Fourier transform on $\{\bf R\}^n$, $n\ge 2$, analyzing the “almost-orthogonality” of the different directions of the space with respect to the Fourier transform. We prove two theorems: the first is related to an angular Littlewood-Paley square function, and we obtain estimates in terms of powers of $\log (N)$, where $N$ is the number of equal angles considered in $\{\bf R\}^2$. The second is an extension of the Hardy-Littlewood maximal function when one consider cylinders of $\{\bf R\}^n$, $n\ge 2$, of fixed eccentricity and direction on a given curve. We obtain sharp estimates for the $L^2$-norm of such operators.},

author = {Cordoba, Antonio},

journal = {Annales de l'institut Fourier},

keywords = {almost-orthogonality; Fourier transform; Hardy-Littlewood maximal function; sharp estimates for the L2-norm},

language = {eng},

number = {3},

pages = {215-226},

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

title = {Geometric Fourier analysis},

url = {http://eudml.org/doc/74547},

volume = {32},

year = {1982},

}

TY - JOUR

AU - Cordoba, Antonio

TI - Geometric Fourier analysis

JO - Annales de l'institut Fourier

PY - 1982

PB - Association des Annales de l'Institut Fourier

VL - 32

IS - 3

SP - 215

EP - 226

AB - In this paper we continue the study of the Fourier transform on ${\bf R}^n$, $n\ge 2$, analyzing the “almost-orthogonality” of the different directions of the space with respect to the Fourier transform. We prove two theorems: the first is related to an angular Littlewood-Paley square function, and we obtain estimates in terms of powers of $\log (N)$, where $N$ is the number of equal angles considered in ${\bf R}^2$. The second is an extension of the Hardy-Littlewood maximal function when one consider cylinders of ${\bf R}^n$, $n\ge 2$, of fixed eccentricity and direction on a given curve. We obtain sharp estimates for the $L^2$-norm of such operators.

LA - eng

KW - almost-orthogonality; Fourier transform; Hardy-Littlewood maximal function; sharp estimates for the L2-norm

UR - http://eudml.org/doc/74547

ER -

## References

top- [1] C. FEFFERMAN, The multiplier problem for the ball, Ann. of Math., 94 (1971). Zbl0234.42009MR45 #5661
- [2] E.M. STEIN and S. WAINGER, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc., 84 (1978). Zbl0393.42010MR80k:42023
- [3] P. TOMAS, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc., 81 (1975). Zbl0298.42011MR50 #10681
- [4] A. CORDOBA, The multiplier problem for the polygon, Ann. of Math., 105 (1977). Zbl0361.42005MR55 #10943
- [5] A. CORDOBA and C. FEFFERMAN, A weighted norm inequality for singular integrals, Studia Math., LVII (1976). Zbl0356.44003MR54 #8132
- [6] S. WAINGER, Averages over low dimensional sets, Proc. Symp. in Pure Math., XXV. Zbl0627.42006

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