Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Second-order elliptic operators and heat kernels on Lie groups

Authors: Ola Bratteli and Derek W. Robinson
Journal: Trans. Amer. Math. Soc. 325 (1991), 683-713
MSC: Primary 22D10; Secondary 35J99, 46L99, 47F05, 58G11
MathSciNet review: 1041043
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Arendt, Batty, and Robinson proved that each second-order strongly elliptic operator $ C$ associated with left translations on the $ {L_p}$-spaces of a Lie group $ G$ generates an interpolating family of semigroups $ T$, whenever the coefficients of $ C$ are sufficiently smooth. We establish that $ T$ has an integral kernel $ K$ satisfying the bounds

$\displaystyle a\prime{t^{ - d/2}}{e^{ - b\prime\vert g{h^{ - 1}}{\vert^2}/t}}{e... ...t}(g;h) \leq a{t^{ - d/2}}{e^{ - b\vert g{h^{ - 1}}{\vert^2}/t}}{e^{\omega t}},$

where $ d$ is the dimension of $ G$, $ \vert g{h^{ - 1}}\vert$ is the right invariant distance from $ h$ to $ g$, and $ a\prime$, $ b\prime$, $ \omega\prime$, etc. are positive constants. Both bounds are derived by generalization of Nash's arguments for pure second-order operators on $ {{\mathbf{R}}^d}$.

References [Enhancements On Off] (What's this?)

  • [ABR] W. Arendt, C. J. K. Batty, and D. W. Robinson, Positive semigroups generated by elliptic operators on Lie groups, J. Operator Theory 23 (1990), 369-408. MR 1066813 (91k:47118)
  • [BGJR] O. Bratteli, F. M. Goodman, P. E. T. Jørgensen, and D. W. Robinson, The heat semigroup and integrability of Lie algebras, J. Funct. Anal. 79 (1988) 351-397. MR 953908 (90a:47105)
  • [Bon] J. M. Bony, Principe du maximum, unicité du problème de Cauchy et inégalité de Harnack pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble) 19 (1969) 277-304. MR 0262881 (41:7486)
  • [BR] O. Bratteli and D. W. Robinson, Operator algebras and quantum statistical mechanics. I, 2nd ed., Springer-Verlag, 1987. MR 611508 (82k:82013)
  • [CFKS] H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger operators, Springer-Verlag, 1987. MR 883643 (88g:35003)
  • [CKS] E. A. Carlin, A. Kusuoko, and D. W. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré Probab. Statist. 2 (1987), 245-287. MR 898496 (88i:35066)
  • [Dav] E. B. Davies, Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math. 109 (1987) 319-333. MR 882426 (88g:58174)
  • [FaS] E. B. Fabes and D. W. Stroock, A new proof of Moser's Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal. 96 (1986) 327-338. MR 855753 (88b:35037)
  • [Går] L. Gårding, Vecteurs analytiques dans des representations des groupes de Lie, Bull. Soc. Math. France 88 (1960), 73-93. MR 0119104 (22:9870)
  • [Kat] T. Kato, Perturbation theory for linear operators, Springer-Verlag, 1966. MR 0203473 (34:3324)
  • [Mos] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964) 101-134. MR 0159139 (28:2357)
  • [Nas] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958) 931-954. MR 0100158 (20:6592)
  • [Rob] D. W. Robinson, Elliptic differential operators on Lie groups, J. Funct. Anal. (to appear). MR 1111188 (92d:22013)
  • [Ste1] E. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956) 482-492. MR 0082586 (18:575d)
  • [Ste2] -, Topics in harmonic analysis, Ann. of Math. Studies, no. 63, Princeton Univ. Press, Princeton, N. J., 1970.
  • [Tre] F. Trèves, Topological vector spaces, distributions, and kernels, Academic Press, 1967. MR 0225131 (37:726)
  • [Var] N. Th. Varopoulos, Analysis on Lie groups, J. Funct. Anal. 76 (1988) 346-410. MR 924464 (89i:22018)
  • [Yos] K. Yosida, Functional analysis, Springer-Verlag, 1974. MR 0350358 (50:2851)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22D10, 35J99, 46L99, 47F05, 58G11

Retrieve articles in all journals with MSC: 22D10, 35J99, 46L99, 47F05, 58G11

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society