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)
Atomic and molecular decompositions of anisotropic Besov spaces. (English) Zbl 1079.42016

In this work the author introduces and studies Besov spaces in R n associated with an expansive dilation A. The starting point is a basic representation formula for tempered distributions

f= q𝒬 f,ϕ Q ψ Q ,where𝒬={A -j ([0,1] n +k):jZ,kZ n },

and ϕ Q and ψ Q are translates and dilates of functions ϕ and ψ that satisfy the Calderón conditions

suppϕ ^,suppψ ^[-π,π] n {0}


jZ ϕ ^((A * ) j ξ) ¯ψ ^((A * ) j ξ)=1forallξR n {0}·

This study extends the isotropic Paley–Littlewood methods of dyadic ϕ–transforms of M. Frazier and B. Jawerth [Indiana Univ. Math J. 34, 777–799 (1985; Zbl 0551.46018); J. Funct. Anal. 93, No. 1, 34–170 (1990; Zbl 0716.46031)] to nonisotropic settings.

42B35Function spaces arising in harmonic analysis
42B25Maximal functions, Littlewood-Paley theory
42C40Wavelets and other special systems
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47B37Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47B38Operators on function spaces (general)
[1]Besov, O.V., Il’in, V.P., Nikol’skii, S.M.: Integral representations of functions and imbedding theorems. Vol. I and II, V. H. Winston & Sons, Washington, D.C., 1979
[2]Bownik, M.: A characterization of affine dual frames in L2(n). Appl. Comput. Harmon. Anal. 8, 203–221 (2000) · Zbl 0961.42018 · doi:10.1006/acha.2000.0284
[3]Bownik, M.: Anisotropic Hardy spaces and wavelets. Mem. Am. Math. Soc. 164(781), 122pp (2003)
[4]Bownik, M., Ho, K.-P.: Atomic and Molecular Decompositions of Anisotropic Triebel-Lizorkin Spaces. Trans. Am. Math. Soc. (to appear)
[5]Buckley, S.M., MacManus, P.: Singular measures and the key of G. Publ. Mat. 44, 483–489 (2000)
[6]Bui, H.-Q.: Weighted Besov and Triebel spaces: interpolation by the real method. Hiroshima Math. J. 12, 581–605 (1982)
[7]Bui, H.-Q.: Characterizations of weighted Besov and Triebel-Lizorkin spaces via temperatures. J. Funct. Anal. 55, 39–62 (1984) · Zbl 0638.46029 · doi:10.1016/0022-1236(84)90017-X
[8]Bui, H.-Q.: Weighted Young’s inequality and convolution theorems on weighted Besov spaces. Math. Nachr. 170, 25–37 (1994) · Zbl 0844.46016 · doi:10.1002/mana.19941700104
[9]Bui, H.-Q., Paluszyński, M., Taibleson, M.H.: A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces. Studia Math. 119, 219–246 (1996)
[10]Bui, H.-Q., Paluszyński, M., Taibleson, M.H.: Characterization of the Besov-Lipschitz and Triebel-Lizorkin spaces. The case q<1. J. Fourier Anal. Appl. 3, 837–846 (1997)
[11]Calderón, A.P., Torchinsky, A.: Parabolic maximal function associated with a distribution. Adv. Math. 16, 1–64 (1975) · Zbl 0315.46037 · doi:10.1016/0001-8708(75)90099-7
[12]Calderón, A.P., Torchinsky, A.: Parabolic maximal function associated with a distribution II. Adv. Math. 24, 101–171 (1977) · Zbl 0355.46021 · doi:10.1016/S0001-8708(77)80016-9
[13]Coifman, R.R.: A real variable characterization of Hp. Studia Math. 51, 269–274 (1974)
[14]Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977) · Zbl 0358.30023 · doi:10.1090/S0002-9904-1977-14325-5
[15]Dintelmann, P.: Classes of Fourier multipliers and Besov-Nikolskij spaces. Math. Nachr. 173, 115–130 (1995) · Zbl 0828.46029 · doi:10.1002/mana.19951730108
[16]Dintelmann, P.: On Fourier multipliers between Besov spaces with 0<p0min {1,p1}. Anal. Math. 22, 113–123 (1996) · Zbl 0906.42007 · doi:10.1007/BF01904968
[17]Farkas, W.: Atomic and subatomic decompositions in anisotropic function spaces. Math. Nachr. 209, 83–113 (2000) · doi:10.1002/(SICI)1522-2616(200001)209:1<83::AID-MANA83>3.0.CO;2-1
[18]Folland, G.B., Stein, E.M.: Hardy spaces on homogeneous groups. Princeton University Press, Princeton, N.J., 1982
[19]Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana U. Math. J. 34, 777–799 (1985) · Zbl 0551.46018 · doi:10.1512/iumj.1985.34.34041
[20]Frazier, M., Jawerth, B.: The φ-transform and applications to distribution spaces. Lecture Notes in Math., 1302, Springer, Berlin Heidelberg, 1988, pp. 223–246
[21]Frazier, M., Jawerth, B.: A Discrete Transform and Decomposition of Distribution Spaces. J. Funct. Anal. 93, 34–170 (1989) · Zbl 0716.46031 · doi:10.1016/0022-1236(90)90137-A
[22]Frazier, M., Jawerth, B., Weiss, G.: Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conference Ser., 79, American Math. Society, 1991
[23]García-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland, 1985
[24]Lemarié-Rieusset, P.-G.: Projecteurs invariants, matrices de dilatation, ondelettes et analyses multi-résolutions. Rev. Mat. Iberoamericana 10, 283–347 (1994)
[25]Lemarié-Rieusset, P.-G.: Recent developments in the Navier-Stokes problem. Chapman & Hall/CRC, 2002
[26]Nazarov, F., Treil, S.: The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis. Algebra i Analiz 8, 32–162 (1996)
[27]Peetre, J.: New thoughts on Besov spaces. Duke University Mathematics Series, No. 1, Mathematics Department, Duke University, Durham, N.C., 1976
[28]Plancherel, M., Pólya, G.: Fonctions entières et intégrales de Fourier multiples. Comment. Math. Helv. 9, 224–248 (1937) · Zbl 0016.36004 · doi:10.1007/BF01258191
[29]Roudenko, S.: Matrix-weighted Besov spaces. Trans. Am. Math. Soc. 355, 273–314 (2003) · Zbl 1010.42011 · doi:10.1090/S0002-9947-02-03096-9
[30]Rychkov, V.S.: On a theorem of Bui, Paluszyński, and Taibleson. Proc. Steklov Inst. Math. 227, 280–292 (1999)
[31]Rychkov, V.S.: Littlewood-Paley theory and function spaces with Alocp weights. Math. Nachr. 224, 145–180 (2001) · doi:10.1002/1522-2616(200104)224:1<145::AID-MANA145>3.0.CO;2-2
[32]Schmeisser, H.-J., Triebel, H.: Topics in Fourier Analysis and Function Spaces. John Wiley & Sons, 1987
[33]Volberg, A.: Matrix Ap weights via S-functions. J. Am. Math. Soc. 10, 445–466 (1997) · Zbl 0877.42003 · doi:10.1090/S0894-0347-97-00233-6
[34]Triebel, H.: Theory of Function Spaces. Monographs in Math., 78, Birkhäuser, Basel, 1983
[35]Triebel, H.: Theory of function spaces II. Monographs in Math., 84, Birkhäuser, Basel, 1992
[36]Triebel, H.: Wavelet bases in anisotropic function spaces. Preprint, 2004