## Anisotropic Triebel-Lizorkin spaces with doubling measures.(English)Zbl 1147.42006

A real $$n\times n$$ matrix is expansive if all of its eigenvalues satisfy $$| \lambda| > 1$$. A quasi-norm associated with an expansive matrix is a Borel measurable mapping $$\rho_A : \mathbb R^n \rightarrow [0,\infty)$$ such that $$\rho_A(x) >0$$ for $$x\not = 0$$, $$\rho_A(Ax) = | \text{det}\,A| \rho_A(x)$$ for $$x\in \mathbb R^n$$, $$\rho_A (x+y) \leq H(\rho_A(x) + \rho_A (y))$$ for $$x, y \in \mathbb R^n$$, where $$H \geq 1$$ is a constant. Put $$B_{\rho_A}(x,r) = \{y \in \mathbb R^n; \rho_A(x-y) < r\}$$, $$x\in\mathbb R^n$$, $$r>0$$. A nonnegative Borel measure is called $$\rho_A$$-doubling if there exists $$\beta = \beta(\mu) > 0$$ such that $\mu(B_{\rho_A} (x,| \text{det}\, A| r)) \leq | \text{det}\,A| ^{\beta} \mu(B_{\rho_A}(x,r)) \;\;\text{for all} \;x\in\mathbb R^n, \;r> 0.$ Let $$\varphi$$ belong to the Schwarz class $$\mathcal S(\mathbb R^n)$$ and satisfy: $\sum_{j\in\mathbb Z} \hat{\varphi} ((A^*)^j\xi) =1 \;\;\text{for all} \;\xi \in \mathbb R^n\setminus \{0\},$ $\text{supp} \;\hat{\varphi} \;\text{is compact and bounded away from the origin}.$ Put $$\varphi_j(x) = | \text{det}\, A| ^j \varphi(A^jx), \;j\in\mathbb Z, \;x\in \mathbb R^n$$.
Given $$\alpha \in\mathbb R$$, $$0<p<\infty$$, $$0<q\leq\infty$$ and a $$\rho_A$$-doubling measure $$\mu$$, the author introduces the anisotropic Triebel-Lizorkin space $$\dot F^{\alpha,q}_p (\mathbb R^n,A,\mu)$$ norm as $\| f\| _{\dot F_p^{\alpha,q}} = \Big\| \Big( \sum_{j\in\mathbb Z} (| \text{det}\, A| ^{j\alpha} | f *\varphi_j| )^q \Big)^{1/q} \Big\| _{L^p(\mu)} < \infty$ and shows that this space is independent of the choice of $$\varphi$$. The operators on $$\dot F^{\alpha,q}_p$$ spaces are studied by transferring them with the use of wavelet transforms to the corresponding sequence spaces $$\dot f^{\alpha,q}_p(A,\mu)$$. In particular, the class of almost diagonal operators is investigated. As an application, smooth atomic and molecular decompositions of spaces $$\dot F^{\alpha,q}_p (A,\mu)$$ are established. The authors also develops localization techniques in the endpoint case $$p=\infty$$. Furthermore, nonsmooth atomic decompositions of spaces $$\dot F^{\alpha,q}_p$$ in the range $$0<p\leq 1$$ are found. Finally, unweighted $$\dot F^{0,2}_p (\mathbb R^n,A)$$ spaces are identified with the anisotropic (real) Hardy spaces $$H^p_A$$ for $$0<p<\infty$$.

### MSC:

 42B25 Maximal functions, Littlewood-Paley theory 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47B38 Linear operators on function spaces (general) 42B35 Function spaces arising in harmonic analysis 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
Full Text:

### References:

  Besov, O. V.; Il’in, V. P.; Nikol’skii, S. M., Integral Representations of Functions and Imbedding Theorems (1979), Washington, DC: I and II, V. H. Winston & Sons, Washington, DC · Zbl 0392.46023  Bownik, M., Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc., 164, 781, 122-122 (2003) · Zbl 1036.42020  Bownik, M., Atomic and molecular decompositions of anisotropic Besov spaces, Math. Z., 250, 539-571 (2005) · Zbl 1079.42016  Bownik, M. Duality and interpolation of anisotropic Triebel-Lizorkin spaces,Math. Z., to appear. · Zbl 1213.42062  Bownik, M.; Ho, K.-P., Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces, Trans. Amer. Math. Soc., 358, 1469-1510 (2006) · Zbl 1083.42016  Buckley, S. M.; MacManus, P., Singular measures and the key ofG, Publ. Mat., 44, 483-489 (2000) · Zbl 0972.60025  Bui, H.-Q., Weighted Besov and Triebel spaces: Interpolation by the real method, Hiroshima Math. J., 12, 581-605 (1982) · Zbl 0525.46023  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) · Zbl 0861.42009  Bui, H.-Q.; Paluszyński, M.; Taibleson, M. H., Characterization of the Besov-Lipschitz and Triebel-Lizorkin spaces. The caseq < 1, J. Fourier Anal. Appl., 3, 837-846 (1997) · Zbl 0897.42010  Calderón, A. P.; Torchinsky, A., Parabolic maximal function associated with a distribution, Adv. in Math., 16, 1-64 (1975) · Zbl 0315.46037  Calderón, A. P.; Torchinsky, A., Parabolic maximal function associated with a distribution II, Adv. in Math., 24, 101-171 (1977) · Zbl 0355.46021  Coifman, R. R., A real variable characterization ofH^p, Studia Math., 51, 269-274 (1974) · Zbl 0289.46037  Coifman, R. R.; Weiss, G., Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83, 569-645 (1977) · Zbl 0358.30023  Parkas, W., Atomic and subatomic decompositions in anisotropic function spaces, Math. Nachr., 209, 83-113 (2000) · Zbl 0954.46021  Fefferman, C.; Stein, E. M., Some maximal inequalities, Amer. J. Math., 91, 107-115 (1971) · Zbl 0222.26019  Fefferman, C.; Stein, E. M., H^p spaces of several variables, Acta Math., 129, 137-193 (1972) · Zbl 0257.46078  Folland, G. B.; Stein, E. M., Hardy Spaces on Homogeneous Groups (1982), Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 0508.42025  Frazier, M.; Jawerth, B., Decomposition of Besov spaces, Indiana Univ. Math. J., 34, 777-799 (1985) · Zbl 0551.46018  Frazier, M. and Jawerth, B. The ϕ-transform and applications to distribution spaces,Lecture Notes in Math.1302, Springer-Verlag, 223-246, (1988). · Zbl 0648.46038  Frazier, M.; Jawerth, B., A Discrete transform and decomposition of distribution spaces, J. Funct. Anal., 93, 34-170 (1990) · Zbl 0716.46031  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). · Zbl 0757.42006  García-Cuerva, J. and Rubio de Francia, J. L.Weighted Norm Inequalities and Related Topics, North- Holland, (1985). · Zbl 0578.46046  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). · Zbl 0994.42009  Grafakos, L.Classical and Modern Fourier Analysis, Pearson Education, (2004). · Zbl 1148.42001  Han, Y. and Sawyer, E. Littlewood-Paley theory on spaces of homogeneous type and classical function spaces,Mem. Amer. Math. Soc.110(530), (1994). · Zbl 0806.42013  Han, Y.; Yang, D., New characterizations and applications of inhomogeneous Besov and Triebel-Lizorkin spaces on homogeneous type spaces and fractals, Dissertationes Math. (Rozprawy Mat.), 403, 102-102 (2002) · Zbl 1019.43006  Han, Y.; Yang, D., Some new spaces of Besov and Triebel-Lizorkin type on homogeneous spaces, Studia Math., 156, 67-97 (2003) · Zbl 1032.42025  Lemarié-Rieusset, P.-G.Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC, (2002). · Zbl 1034.35093  Meyer, Y., Wavelets and Operators (1992), Cambridge: Cambridge University Press, Cambridge · Zbl 0776.42019  Rychkov, V. S., Littlewood-Paley theory and function spaces with A_p^loc weights, Math. Nachr., 224, 145-180 (2001) · Zbl 0984.42011  Stein, E. M.Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, (1993). · Zbl 0821.42001  Schmeisser, H.-J. and Triebel, H.Topics in Fourier Analysis and Function Spaces, John Wiley & Sons, (1987). · Zbl 0661.46024  Triebel, H. Theory of function spaces,Monogr. Math.78, Birkhäuser, (1983). · Zbl 0546.46028  Triebel, H., Theory of function Spaces II, Monogr. Math. (1992), Basel: Birkhäuser Verlag, Basel · Zbl 0763.46025  Triebel, H. Wavelet bases in anisotropic function spaces,Function Spaces, Differential Operators and Nonlinear Analysis, 370-387, (2004).  Triebel, H., Theory of function spaces III, Monogr. Math. (2006), Basel: Birkhäuser Verlag, Basel · Zbl 1104.46001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.