Estimates of global bounds for some Schrödinger heat kernels on manifolds

Illinois Journal of Mathematics

Volumn 44, Issue 3, 2000, Pages 556-573

  Zhang, Qi S ^a   Zhao, Z ^b  

^a University of Memphis   (United States)

^b UNIVERSITY OF MISSOURI   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034259049     PISSN: 00192082     EISSN: None     Source Type: Journal    
DOI: 10.1215/ijm/1256060416     Document Type: Article

References (25)

1. D. G. Aronson, Non-negative solutions of linear parabolic equations, Annali della Scuola Norm. Sup. Pisa 22 (1968), 607-694.
2. M. Aizenman and B. Simon, Brownian motion and Harnack's inequality for Schrödinger operators, Comm. Pure Appl. Math 35 (1982), 209-271.
3. P. Baras and J. A. Goldstein, The heat equation with a singular potential, Tran. Amer. Math. Soc. 284 (1984), 121-139.
4. K. L. Chung and Z. Zhao, From Brownian motion to Schrödinger equations, Springer, Berlin, 1995, pp. 415-425.
5. E. B. Fabes and D. W. Stroock, A new proof of Moser's parabolic Harnack inequality using the old idea of Nash, Arch. Rational Mech. Anal. 96 (1986), 327-338.
6. A. Grigor'yan and W. Hansen, A Liouville property for Schrödinger operators. Math. Ann. 312 (1998), 659-716.
7. p norms of non-critical Schrödinger operators, J. Funct. Anal. 102 (1991), 95-115.
8. F. Gesztesy and Z. Zhao, On positive solutions of critical Schrödinger operators in two dimensions, J. Funct. Anal. 127 (1995), 235-256.
9. P. Li, Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature, Ann. Math. 124 (1986), 1-21.
10. P. Li and L-F. Tam, Green's functions, harmonic functions and volume comparison, J. Differential Geom. 41 (1995), 277-318.
11. P. Li and S. T. Yau, On the parabolic kernel of Schrödinger operators, Acta Math. 156 (1986), 153-201.
12. M. Murata, Positive solutions and large time behavior of of Schrödinger semigroups, Simon's problem. J. Funct. Anal. 56 (1984), 300-310.
13. J. Norris, Long-time behavior of heat flow, global estimates and exact asymptotics, Arch. Rational Mech. Anal. 140 (1997), 161-195.
14. J. Norris and D. Stroock, Estimates on the fundamental solution to the heat flow with uniformly elliptic coefficients, Proc. London Math. Soc. 62 (1991), 373-402.
15. p-spectrum of generalized Schrödinger operators and equivalence of Green's functions, Int. Math. Research Notices (1997), 573-593.
16. p properties of Schrödinger operators and localization of binding, J. Funct. Anal. 35 (1980), 215-229.
17. p norm of Schrödinger semigroups, J. Funct. Anal. 40 (1981), 66-83.
18. _, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 447-526.
19. A. Sa Barreto and M. Zworski, Existence of resonance in potential scattering, Comm. PAM XLIX (1996), 1271-1280.
20. Qi S. Zhang, An optimal parabolic estimate and its applications in prescribing scalar curvature on some open manifolds with Ricci ≥ 0, Math. Ann. 316 (2000), 703-731.
21. _, On a parabolic equation with a singular lower order term. Part II, Indiana Univ. Math. J. 46 (1997), 989-1020.
22. _, Semilinear parabolic problems on manifolds and applications to the non-compact Yamabe problem, 1997, preprint.
23. Z. Zhao, On the existence of positive solutions of nonlinear elliptic equations-A probabilistic potential theory approach, Duke Math J. 69 (1993), 247-258.
24. -, Subcriticality and guageability for the Schrödinger operator, Trans. Amer. Math. Soc. 334 (1992), 75-96.
25. _, On bounded positive solutions and generalized Liouville theorem for Schrödinger equations, preprint.