Riesz potentials and amalgams

Michael Cowling; Stefano Meda; Roberta Pasquale

Annales de l'institut Fourier (1999)

  • Volume: 49, Issue: 4, page 1345-1367
  • ISSN: 0373-0956

Abstract

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Let ( M , d ) be a metric space, equipped with a Borel measure μ satisfying suitable compatibility conditions. An amalgam A p q ( M ) is a space which looks locally like L p ( M ) but globally like L q ( M ) . We consider the case where the measure μ ( B ( x , ρ ) of the ball B ( x , ρ ) with centre x and radius ρ behaves like a polynomial in ρ , and consider the mapping properties between amalgams of kernel operators where the kernel ker K ( x , y ) behaves like d ( x , y ) - a when d ( x , y ) ≤ 1 and like d ( x , y ) - b when d ( x , y ) ≥ 1 . As an application, we describe Hardy–Littlewood–Sobolev type regularity theorems for Laplace–Beltrami operators on Riemannian manifolds and for certain subelliptic operators on Lie groups of polynomial growth.

How to cite

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Cowling, Michael, Meda, Stefano, and Pasquale, Roberta. "Riesz potentials and amalgams." Annales de l'institut Fourier 49.4 (1999): 1345-1367. <http://eudml.org/doc/75384>.

@article{Cowling1999,
abstract = {Let $(M,d)$ be a metric space, equipped with a Borel measure $\mu $ satisfying suitable compatibility conditions. An amalgam $A^q_p (M) $ is a space which looks locally like $L^p(M)$ but globally like $L^q (M)$. We consider the case where the measure $\mu (B(x,\rho )$ of the ball $B(x,\rho )$ with centre $x$ and radius $\rho $ behaves like a polynomial in $\rho $, and consider the mapping properties between amalgams of kernel operators where the kernel $\ker \,K (x,y)$ behaves like $d(x,y)^\{-a\}$ when $d(x,y) \le 1$ and like $d(x,y)^\{-b\}$ when $d(x,y) \ge 1$. As an application, we describe Hardy–Littlewood–Sobolev type regularity theorems for Laplace–Beltrami operators on Riemannian manifolds and for certain subelliptic operators on Lie groups of polynomial growth.},
author = {Cowling, Michael, Meda, Stefano, Pasquale, Roberta},
journal = {Annales de l'institut Fourier},
keywords = {amalgam; heat equation; Gaussian semigroup; polynomial growth; Riesz potentials; infinite cylinder; Laplace-Beltrami operator; Hardy-Littlewood-Sobolev regularity theorem},
language = {eng},
number = {4},
pages = {1345-1367},
publisher = {Association des Annales de l'Institut Fourier},
title = {Riesz potentials and amalgams},
url = {http://eudml.org/doc/75384},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Cowling, Michael
AU - Meda, Stefano
AU - Pasquale, Roberta
TI - Riesz potentials and amalgams
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 4
SP - 1345
EP - 1367
AB - Let $(M,d)$ be a metric space, equipped with a Borel measure $\mu $ satisfying suitable compatibility conditions. An amalgam $A^q_p (M) $ is a space which looks locally like $L^p(M)$ but globally like $L^q (M)$. We consider the case where the measure $\mu (B(x,\rho )$ of the ball $B(x,\rho )$ with centre $x$ and radius $\rho $ behaves like a polynomial in $\rho $, and consider the mapping properties between amalgams of kernel operators where the kernel $\ker \,K (x,y)$ behaves like $d(x,y)^{-a}$ when $d(x,y) \le 1$ and like $d(x,y)^{-b}$ when $d(x,y) \ge 1$. As an application, we describe Hardy–Littlewood–Sobolev type regularity theorems for Laplace–Beltrami operators on Riemannian manifolds and for certain subelliptic operators on Lie groups of polynomial growth.
LA - eng
KW - amalgam; heat equation; Gaussian semigroup; polynomial growth; Riesz potentials; infinite cylinder; Laplace-Beltrami operator; Hardy-Littlewood-Sobolev regularity theorem
UR - http://eudml.org/doc/75384
ER -

References

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  1. [1] J.-P. BERTRANDIAS, C. DATRY and C. DUPUIS, Unions et intersections d'espaces Lp invariantes par translation ou convolution, Ann. Inst. Fourier, Grenoble, 28-2 (1978), 53-84. Zbl0365.46029MR81g:43005
  2. [2] R.L. BISHOP and R.J. CRITTENDEN, Geometry of Manifolds, Academic Press, New York, 1964. Zbl0132.16003
  3. [3] C. CARATHÉODORY, Untersuchungen über dire Grundlagen der Thermodynamik, Math. Ann., 67 (1909), 355-386. 
  4. [4] J. CHEEGER, M. GROMOV and M. TAYLOR, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom., 17 (1982), 15-53. Zbl0493.53035MR84b:58109
  5. [5] W.L. CHOW, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., 117 (1940), 98-115. Zbl0022.02304MR1,313dJFM65.0398.01
  6. [6] T. COULHON, Dimension à l'infini d'un semi-groupe analytique, Bull. Sci. Math., 114 (1990), 485-500. Zbl0738.47032MR91k:47091
  7. [7] T. COULHON, Noyau de la chaleur et discrétisation d'une variété riemannienne, Israel J. Math., 80 (1992), 289-300. Zbl0772.58055MR93k:58213
  8. [8] T. COULHON, L. SALOFF-COSTE, Variétés riemanniennes isométriques à l'infini, Rev. Mat. Iberoamericana, 11 (1995), 687-726. Zbl0845.58054MR96m:53035
  9. [9] T. COULHON, L. SALOFF-COSTE, Semi-groupes d'opérateurs et espaces fonctionnels sur les groupes de Lie, J. Approx. Theory, 65 (1991), 176-199. Zbl0745.47030MR92c:47049
  10. [10] C.B. CROKE, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. Ecole Norm. Sup., 13 (1980), 419-435. Zbl0465.53032MR83d:58068
  11. [11] E.B. DAVIES, Heat Kernels and Spectral Theory, Cambridge Tract in Math. 92, Cambridge University Press, Cambridge, 1989. Zbl0699.35006MR90e:35123
  12. [12] E.B. DAVIES, Gaussian upper bounds for the heat kernels of some second order operators on Riemannian manifolds, J. Funct. Anal., 80 (1988), 16-32. Zbl0759.58045MR90k:58213
  13. [13] E.B. DAVIES and M.M.H. PANG, Sharp heat bounds for some Laplace operators, Quart. J. Math. Oxford, 40 (1989), 281-290. Zbl0701.35004MR91i:58142
  14. [14] G.B. FOLLAND and E.M. STEIN, Estimates for the ∂b-complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 27 (1974), 429-522. Zbl0293.35012MR51 #3719
  15. [15] J.J.F. FOURNIER and J. STEWART, Amalgams of Lp and lq, Bull. Amer. Math. Soc. (N.S.), 13 (1985), 1-21. Zbl0593.43005MR86f:46027
  16. [16] F. HOLLAND, Harmonic analysis on amalgams of Lp and lq, J. London Math. Soc., 10 (1975), 295-305. Zbl0314.46029MR51 #11013
  17. [17] L. HÖRMANDER, Hypoelliptic second-order differential equations, Acta Math., 119 (1967), 147-171. Zbl0156.10701MR36 #5526
  18. [18] M. KANAI, Rough isometries, and combinatorial approximation of non-compact Riemannian manifolds, J. Math. Soc. Japan, 37 (1985), 391-413. Zbl0554.53030MR87d:53082
  19. [19] P. LI and S.T. YAU, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201. Zbl0611.58045MR87f:58156
  20. [20] A. NAGEL, E.M. STEIN and S. WAINGER, Balls and metrics defined by vector fields. I: Basic properties, Acta Math., 155 (1985), 103-147. Zbl0578.32044MR86k:46049
  21. [21] D.W. ROBINSON, Elliptic Operators and Lie Groups, Oxford Mathematical Monographs, Oxford University Press, Oxford, 1991. Zbl0747.47030
  22. [22] L. SALOFF-COSTE, Uniformly elliptic operators on Riemannian manifolds, J. Differential Geom., 36 (1992), 417-450. Zbl0735.58032MR93m:58122
  23. [23] N.Th. VAROPOULOS, Analysis on Lie groups, J. Funct. Anal., 76 (1988), 346-410. Zbl0634.22008MR89i:22018
  24. [24] N.Th. VAROPULOS, L. SALOFF-COSTE and T. COULHON, Analysis and Geometry on Groups, Cambridge Tract in Math. 100, Cambridge University Press, Cambridge, 1992. Zbl0813.22003
  25. [25] S.-T. YAU, Some function theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J., 25 (1976), 659-670. Zbl0335.53041MR54 #5502

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