zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Anisotropic Triebel-Lizorkin spaces with doubling measures. (English) Zbl 1147.42006

A real n×n matrix is expansive if all of its eigenvalues satisfy |λ|>1. A quasi-norm associated with an expansive matrix is a Borel measurable mapping ρ A : n [0,) such that ρ A (x)>0 for x0, ρ A (Ax)=|detA|ρ A (x) for x n , ρ A (x+y)H(ρ A (x)+ρ A (y)) for x,y n , where H1 is a constant. Put B ρ A (x,r)={y n ;ρ A (x-y)<r}, x n , r>0. A nonnegative Borel measure is called ρ A -doubling if there exists β=β(μ)>0 such that

μ(B ρ A (x,|detA|r))|detA| β μ(B ρ A (x,r))forallx n ,r>0·

Let ϕ belong to the Schwarz class 𝒮( n ) and satisfy:

j ϕ ^((A * ) j ξ)=1forallξ n {0},
suppϕ ^iscompactandboundedawayfromtheorigin·

Put ϕ j (x)=|detA| j ϕ(A j x),j,x n .

Given α, 0<p<, 0<q and a ρ A -doubling measure μ, the author introduces the anisotropic Triebel-Lizorkin space F ˙ p α,q ( n ,A,μ) norm as

f F ˙ p α,q = j (|detA| jα |f* ϕ j |) q 1/q L p (μ) <

and shows that this space is independent of the choice of ϕ. The operators on F ˙ p α,q spaces are studied by transferring them with the use of wavelet transforms to the corresponding sequence spaces f ˙ p α,q (A,μ). In particular, the class of almost diagonal operators is investigated. As an application, smooth atomic and molecular decompositions of spaces F ˙ p α,q (A,μ) are established. The authors also develops localization techniques in the endpoint case p=. Furthermore, nonsmooth atomic decompositions of spaces F ˙ p α,q in the range 0<p1 are found. Finally, unweighted F ˙ p 0,2 ( n ,A) spaces are identified with the anisotropic (real) Hardy spaces H A p for 0<p<.


MSC:
42B25Maximal functions, Littlewood-Paley theory
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)
42B35Function spaces arising in harmonic analysis
42C40Wavelets and other special systems
References:
[1]Besov, O. V., Il’in, V. P., and Nikol’skii, S. M.Integral Representations of Functions and Imbedding Theorems, I and II, V. H. Winston &amp; Sons, Washington, DC, (1979).
[2]Bownik, M. Anisotropic Hardy spaces and wavelets,Mem. Amer. Math. Soc. 164(781), 122, (2003).
[3]Bownik, M. Atomic and molecular decompositions of anisotropic Besov spaces,Math. Z. 250, 539–571, (2005). · Zbl 1079.42016 · doi:10.1007/s00209-005-0765-1
[4]Bownik, M. Duality and interpolation of anisotropic Triebel-Lizorkin spaces,Math. Z., to appear.
[5]Bownik, M. and Ho, K.-P. Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces,Trans. Amer. Math. Soc. 358, 1469–1510, (2006). · Zbl 1083.42016 · doi:10.1090/S0002-9947-05-03660-3
[6]Buckley, S. M. and MacManus, P. Singular measures and the key ofG, Publ. Mat. 44, 483–489, (2000). · Zbl 0972.60025 · doi:10.5565/PUBLMAT_44200_07
[7]Bui, H.-Q. Weighted Besov and Triebel spaces: Interpolation by the real method,Hiroshima Math. J. 12, 581–605, (1982).
[8]Bui, H.-Q., Paluszyński, M., and Taibleson, M. H. A maximal function characterization of weighted Besov- Lipschitz and Triebel-Lizorkin spaces,Studia Math. 119, 219–246, (1996).
[9]Bui, H.-Q., Paluszyński, M., and Taibleson, M. H. Characterization of the Besov-Lipschitz and Triebel-Lizorkin spaces. The caseq &lt; 1,J. Fourier Anal. Appl. 3(Special Issue), 837–846, (1997). · Zbl 0897.42010 · doi:10.1007/BF02656489
[10]Calderón, A. P. and Torchinsky, A. Parabolic maximal function associated with a distribution,Adv. in Math. 16, 1–64, (1975). · Zbl 0315.46037 · doi:10.1016/0001-8708(75)90099-7
[11]Calderón, A. P. and Torchinsky, A. Parabolic maximal function associated with a distribution II,Adv. in Math. 24, 101–171, (1977). · Zbl 0355.46021 · doi:10.1016/S0001-8708(77)80016-9
[12]Coifman, R. R. A real variable characterization ofH p,Studia Math. 51, 269–274, (1974).
[13]Coifman, R. R. and Weiss, G. Extensions of Hardy spaces and their use in analysis,Bull. Amer. Math. Soc. 83, 569–645, (1977). · Zbl 0358.30023 · doi:10.1090/S0002-9904-1977-14325-5
[14]Parkas, 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
[15]Fefferman, C. and Stein, E. M. Some maximal inequalities,Amer. J. Math. 91, 107–115, (1971). · Zbl 0222.26019 · doi:10.2307/2373450
[16]Fefferman, C. and Stein, E. M.H p spaces of several variables,Acta Math. 129, 137–193, (1972). · Zbl 0257.46078 · doi:10.1007/BF02392215
[17]Folland, G. B. and Stein, E. M.Hardy Spaces on Homogeneous Groups, Princeton University Press, Princeton, NJ, (1982).
[18]Frazier, M. and Jawerth, B. Decomposition of Besov spaces,Indiana Univ. Math. J. 34, 777–799, (1985). · Zbl 0551.46018 · doi:10.1512/iumj.1985.34.34041
[19]Frazier, M. and Jawerth, B. The -transform and applications to distribution spaces,Lecture Notes in Math. 1302, Springer-Verlag, 223–246, (1988).
[20]Frazier, M. and Jawerth, B. A Discrete transform and decomposition of distribution spaces,J. Funct. Anal. 93, 34–170, (1990). · Zbl 0716.46031 · doi:10.1016/0022-1236(90)90137-A
[21]Frazier, M., Jawerth, B., and Weiss, G. Littlewood-Paley theory and the study of function spaces,CBMS Reg. Conf. Ser. Math. 79, American Math. Society, (1991).
[22]García-Cuerva, J. and Rubio de Francia, J. L.Weighted Norm Inequalities and Related Topics, North- Holland, (1985).
[23]Gilbert, J., Han, Y., Hogan, J., Lakey, J., Weiland, D., and Weiss, G. Smooth molecular decompositions of functions and singular integral operators,Mem. Amer. Math. Soc. 156, (2002).
[24]Grafakos, L.Classical and Modern Fourier Analysis, Pearson Education, (2004).
[25]Han, Y. and Sawyer, E. Littlewood-Paley theory on spaces of homogeneous type and classical function spaces,Mem. Amer. Math. Soc. 110(530), (1994).
[26]Han, Y. and Yang, D. New characterizations and applications of inhomogeneous Besov and Triebel-Lizorkin spaces on homogeneous type spaces and fractals,Dissertationes Math. (Rozprawy Mat.) 403, 102, (2002).
[27]Han, Y. and Yang, D. Some new spaces of Besov and Triebel-Lizorkin type on homogeneous spaces,Studia Math. 156, 67–97, (2003). · Zbl 1032.42025 · doi:10.4064/sm156-1-5
[28]Lemarié-Rieusset, P.-G.Recent Developments in the Navier-Stokes Problem, Chapman &amp; Hall/CRC, (2002).
[29]Meyer, Y.Wavelets and Operators, Cambridge University Press, Cambridge, (1992).
[30]Rychkov, V. S. Littlewood-Paley theory and function spaces with A p loc weights,Math. Nachr. 224, 145–180, (2001). · doi:10.1002/1522-2616(200104)224:1<145::AID-MANA145>3.0.CO;2-2
[31]Stein, E. M.Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, (1993).
[32]Schmeisser, H.-J. and Triebel, H.Topics in Fourier Analysis and Function Spaces, John Wiley &amp; Sons, (1987).
[33]Triebel, H. Theory of function spaces,Monogr. Math. 78, Birkhäuser, (1983).
[34]Triebel, H. Theory of function Spaces II,Monogr. Math. 84, Birkhäuser Verlag, Basel, (1992).
[35]Triebel, H. Wavelet bases in anisotropic function spaces,Function Spaces, Differential Operators and Nonlinear Analysis, 370–387, (2004).
[36]Triebel, H. Theory of function spaces III,Monogr. Math. 100, Birkhäuser Verlag, Basel, (2006).