zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Linear and bilinear multiplier operators for the Dunkl transform. (English) Zbl 1209.42006
The paper treats the problem of L p estimates for linear and bilinear multiplier operators for the Dunkl transform in the one dimensional case. Using Hörmander’s technique and an explicit formula for the Dunkl translation operator, an analogue of the celebrated Hörmander multiplier theorem is proved in the Dunkl setting. In the bilinear case a strategy of Coifman and Meyer is applied to receive an L p estimate for the bilinear multiplier operators.
42B15Multipliers, several variables
42B20Singular and oscillatory integrals, several variables
46F12Integral transforms in distribution spaces
[1]J.-Ph. Anker, L p Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann. of Math. (2) 132 (1990), no. 3, 597-628.
[2]J. J. Betancor, Ò. Ciaurri, and J. L. Varona, The multiplier of the interval [-1, 1] for the Dunkl transform on the real line, J. Funct. Anal. 242 (2007), 327-336.
[3]R. Coifman, and Y. Meyer, Commutateurs d’intégrales singulières et opérateurs multilinéaires, Ann. Inst. Fourier, Grenoble 28 (1978) no. 3, 177-202.
[4]R. Coifman, and Y. Meyer, Opérateurs multilinéaires, Hermann, Paris, [1991].
[5]Dunkl C.: Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311, 167–183 (1989) · doi:10.1090/S0002-9947-1989-0951883-8
[6]Dunkl C.: Hankel transforms associated to finite reflection groups. Contemp. Math. 138, 123–138 (1992)
[7]M. F. E. de Jeu, The Dunkl transform. Invent. Math. 113 (1993), 147-162. · Zbl 0789.33007 · doi:10.1007/BF01244305
[8]J. Gosselin and K. Stempak, A weak-type estimate for Fourier-Bessel multipliers, Proc. Amer. Math. Soc., 106 (1989), no. 3, 655-662.
[9]Haagerup U.: The best constants in the Khintchine inequality. Studia Math. 70, 231–283 (1981)
[10]L. Hörmander, Estimates for translation invariant operators on L p spaces, Acta Math., 104 (1960) 93-140. · Zbl 0093.11402 · doi:10.1007/BF02547187
[11]A. Nowak and K. Stempak, Relating transplantation and multiplier for Dunkl and Hankel transforms, Math. Nachr. 281 (2008), no. 11, 1604-1611.
[12]M. Rösler, Bessel-type signed hypergroups on , Probability measures on groups and related structures, XI(Oberwolfach, 1994), 292-304, World Sci., Publ., River Edge, NJ, 1995.
[13]Soltani F.: L p -Fourier multipliers for the Dunkl operators on the real line. J. Funct. Anal., 209, 16–35 (2004) · Zbl 1045.43003 · doi:10.1016/j.jfa.2003.11.009
[14]E. M. Stein, Harmonic Analysis: Reals-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton, New Jersey 1993.
[15]S. Thangavelu and Y. Xu, Convolution operator and maximal function for the Dunkl transform, Journal d’analyse mathématique. 97 (2005), no.1, 25-55.
[16]Trimèche K.: Paley-Wiener Theorem for the Dunkl transform and Dunkl translation. Integral Transforms Spec. Funct. 13, 17–38 (2002) · Zbl 1030.44004 · doi:10.1080/10652460212888