Besov spaces on symmetric manifolds—the atomic decomposition

• Autorzy: Leszek Skrzypczak
• Języki publikacji: EN

Abstrakty

EN

We give the atomic decomposition of the inhomogeneous Besov spaces defined on symmetric Riemannian spaces of noncompact type. As an application we get a theorem of Bernstein type for the Helgason-Fourier transform.

Kategorie tematyczne

• 46E35: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
• 43A85: Analysis on homogeneous spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1997
• Tom: 124
• Numer: 3
• Strony: 215-238

Daty

wydano

1997

otrzymano

1995-01-26

poprawiono

1996-12-12

Twórcy

autor

Leszek Skrzypczak

• Faculty of Mathematics and Computer Science, A. Mickiewicz University, Matejki 48/49 60-769 Poznań, Poland

Bibliografia

• [1] J.-P. Anker, $L_p$-Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann. of Math. 132 (1990), 597-628.
• [2] J.-P. Anker, The spherical Fourier transform of rapidly decreasing functions. A simple proof of a characterization due to Harish-Chandra, Helgason, Trombi, and Varadarajan, J. Funct. Anal. 96 (1991), 331-349.
• [3] H. Q. Bui, Representation theorems and atomic decomposition of Besov spaces, Math. Nachr. 132 (1987), 301-311.
• [4] M. Eguchi, Asymptotic expansions of Eisenstein integrals and Fourier transform on symmetric spaces, J. Funct. Anal. 34 (1979), 164-216.
• [5] H. G. Feichtinger and K. Gröchenig, A unified approach to atomic decompositions via integrable group representations, in: Function Spaces and Applications, Proc. Conf. Lund 1986, Lecture Notes in Math. 1302, Springer, 1988, 52-73.
• [6] H. G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, I, J. Funct. Anal. 86 (1989), 307-340; II, Monatsh. Math. 108 (1989), 129-148.
• [7] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777-799.
• [8] M. Frazier and B. Jawerth, A discrete transform and decomposition of distribution spaces, J. Funct. Anal. 93 (1990), 34-170.
• [9] M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley Theory and Study of Function Spaces, CBMS Regional Conf. Ser. in Math. 79, Amer. Math. Soc., 1991.
• [10] R. Gangolli and V. S. Varadarajan, Harmonic Analysis of Spherical Functions on Real Reductive Groups, Ergeb. Math. Grenzgeb. 101, Springer, 1988.
• [11] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, 1978.
• [12] S. Helgason, Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions, Academic Press, 1984.
• [13] S. Helgason, The surjectivity of invariant differential operators on symmetric spaces I, Ann. of Math. 98 (1973), 451-480.
• [14] T. Kawazoe, Atomic Hardy spaces on semisimple Lie groups, in: Non-Commutative Harmonic Analysis and Lie Groups, Proc. Conf. Marseille 1985, Lecture Notes in Math. 1243, Springer, 1987, 189-197.
• [15] R. A. Macías and C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. in Math. 33 (1979), 271-309.
• [16] L. Skrzypczak, Some equivalent norms in Sobolev and Besov spaces on symmetric manifolds, J. London Math. Soc. 53 (1996), 569-581.
• [17] L. Skrzypczak, Vector-valued Fourier multipliers on symmetric spaces of the noncompact type, Monatsh. Math. 119 (1995), 99-123.
• [18] H. Triebel, Atomic decomposition of $F^s_p,q$ spaces. Applications to exotic pseudodifferential and Fourier integral operators, Math. Nachr. 144 (1989), 189-222.
• [19] H. Triebel, How to measure smoothness of distributions on Riemannian symmetric manifolds and Lie groups?, Z. Anal. Anwendungen 7 (1988), 471-480.
• [20] H. Triebel, Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds, Ark. Mat. 24 (1986), 300-337.
• [21] H. Triebel, Theory of Function Spaces II, Birkhäuser, 1992.