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Volumn 194, Issue 1, 2002, Pages 53-141

Martin boundaries of elliptic skew products, semismall perturbations, and fundamental solutions of parabolic equations

Author keywords

Elliptic equation; Fundamental solution; Green function; Intrinsic ultracontractivity; Martin boundary; Parabolic equation; Positive solution; Semismall perturbation; Skew product

Indexed keywords


EID: 0037057030     PISSN: 00221236     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-1236(01)93939-4     Document Type: Article
Times cited : (38)

References (98)
  • 1
    • 0001786125 scopus 로고
    • On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds
    • (D. Greco, Ed.) Liguori Editore, Naples
    • S. Agmon, On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds, in "Methods of Functional Analysis and Theory of Elliptic Equations" (D. Greco, Ed.), pp. 19-52, Liguori Editore, Naples, 1982.
    • (1982) "Methods of Functional Analysis and Theory of Elliptic Equations" , pp. 19-52
    • Agmon, S.1
  • 2
    • 0032329262 scopus 로고    scopus 로고
    • Norm estimate of Green operator, perturbation of Green function and integrability of superharmonic functions
    • H. Aikawa, Norm estimate of Green operator, perturbation of Green function and integrability of superharmonic functions, Math. Ann. 312 (1998), 289-318.
    • (1998) Math. Ann , vol.312 , pp. 289-318
    • Aikawa, H.1
  • 3
    • 0035219587 scopus 로고    scopus 로고
    • Boundary Harnack principle and Martin boundary for a uniform domain
    • H. Aikawa, Boundary Harnack principle and Martin boundary for a uniform domain, J. Math. Soc. Japan 53 (2001), 119-145.
    • (2001) J. Math. Soc. Japan , vol.53 , pp. 119-145
    • Aikawa, H.1
  • 4
    • 0000801374 scopus 로고    scopus 로고
    • Generalized Cranston-McConnell inequalities and Martin boundaries of unbounded domains
    • H. Aikawa and M. Murata, Generalized Cranston-McConnell inequalities and Martin boundaries of unbounded domains, J. Analyse Math. 69 (1996), 137-152.
    • (1996) J. Analyse Math , vol.69 , pp. 137-152
    • Aikawa, H.1    Murata, M.2
  • 5
    • 0001577515 scopus 로고
    • Principe de Harnack à frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien
    • A. Ancona, Principe de Harnack à frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien, Ann. Inst. Fourier (Grenoble) 28 (1978), 169-213.
    • (1978) Ann. Inst. Fourier (Grenoble) , vol.28 , pp. 169-213
    • Ancona, A.1
  • 6
    • 0000126969 scopus 로고
    • Negatively curved manifolds, elliptic operators and the Martin boundary
    • A. Ancona, Negatively curved manifolds, elliptic operators and the Martin boundary, Ann. Math. 121 (1987), 429-461.
    • (1987) Ann. Math , vol.121 , pp. 429-461
    • Ancona, A.1
  • 7
    • 0001591776 scopus 로고
    • "Theorie du potential sur les graphes et les varietes"
    • Springer, Berlin
    • A. Ancona, "Theorie du potential sur les graphes et les varietes," Lecture Notes in Mathematics, Vol. 1427, pp. 1-23, Springer, Berlin, 1988.
    • (1988) Lecture Notes in Mathematics , vol.1427 , pp. 1-23
    • Ancona, A.1
  • 8
    • 0002417396 scopus 로고    scopus 로고
    • First eigenvalues and comparison of Green's functions for elliptic operators on manifolds or domains
    • A. Ancona, First eigenvalues and comparison of Green's functions for elliptic operators on manifolds or domains, J. Analyse Math. 72 (1997), 45-92.
    • (1997) J. Analyse Math , vol.72 , pp. 45-92
    • Ancona, A.1
  • 9
    • 0010864043 scopus 로고    scopus 로고
    • Some results and examples about the behavior of harmonic functions and Green's functions with respect to second order elliptic operators
    • Preprint, Univ. Paris Sud
    • A. Ancona, Some results and examples about the behavior of harmonic functions and Green's functions with respect to second order elliptic operators, Preprint, Univ. Paris Sud., 1999.
    • (1999)
    • Ancona, A.1
  • 10
    • 0001550136 scopus 로고
    • Positive harmonic functions on complete manifolds of negative curvature
    • M. T. Anderson and R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. Math. 121 (1985), 429-461.
    • (1985) Ann. Math , vol.121 , pp. 429-461
    • Anderson, M.T.1    Schoen, R.2
  • 11
    • 0000377165 scopus 로고
    • Non-negative solutions of linear parabolic equations
    • D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Sci. Norm. Sup. Pisa 22 (1968), 607-694.
    • (1968) Ann. Sci. Norm. Sup. Pisa , vol.22 , pp. 607-694
    • Aronson, D.G.1
  • 12
    • 0003131889 scopus 로고
    • Local behavior of solutions of quasilinear parabolic equations
    • D. G. Aronson, and J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal. 25 (1967), 81-122.
    • (1967) Arch. Rational Mech. Anal , vol.25 , pp. 81-122
    • Aronson, D.G.1    Serrin, J.2
  • 13
    • 0001488896 scopus 로고
    • Théorie du renouvellement pour des chaînes semi-markoviennes transientes
    • M. Babillot, Théorie du renouvellement pour des chaînes semi-markoviennes transientes, Ann. Inst. H. Poincaré 24 (1988), 507-569.
    • (1988) Ann. Inst. H. Poincaré , vol.24 , pp. 507-569
    • Babillot, M.1
  • 14
    • 0001743189 scopus 로고
    • Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger operators
    • R. Bañuelos, Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger operators, J. Funct. Anal. 100 (1991), 181-206.
    • (1991) J. Funct. Anal , vol.100 , pp. 181-206
    • Bañuelos, R.1
  • 15
  • 16
    • 85031388096 scopus 로고
    • Singularités de certains champ scalaires
    • G. Bouligand, Singularités de certains champ scalaires, Ann. Sci. École Norm. Sup. 48 (1931), 95-152.
    • (1931) Ann. Sci. École Norm. Sup , vol.48 , pp. 95-152
    • Bouligand, G.1
  • 18
    • 0000736848 scopus 로고
    • Boundary behavior of nonnegative solutions of elliptic operators in divergence form
    • L. A. Caffarelli, E. Fabes, S. Mortola, and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J. 30 (1981), 621-640.
    • (1981) Indiana Univ. Math. J , vol.30 , pp. 621-640
    • Caffarelli, L.A.1    Fabes, E.2    Mortola, S.3    Salsa, S.4
  • 19
    • 0001763939 scopus 로고
    • Harnack's inequality and mean-value inequalities for solutions of degenerate elliptic equations
    • S. Chanillo and R. L. Wheeden, Harnack's inequality and mean-value inequalities for solutions of degenerate elliptic equations, Comm. Partial Differential Equations 11 (1986), 1111-1134.
    • (1986) Comm. Partial Differential Equations , vol.11 , pp. 1111-1134
    • Chanillo, S.1    Wheeden, R.L.2
  • 20
    • 0010072199 scopus 로고
    • Degenerate parabolic equations and Harnack inequality
    • F. Chiarenza and R. Serapioni, Degenerate parabolic equations and Harnack inequality, Ann. Mat. Pura Appl. 4 (1983), 139-162.
    • (1983) Ann. Mat. Pura Appl , vol.4 , pp. 139-162
    • Chiarenza, F.1    Serapioni, R.2
  • 22
    • 0001217124 scopus 로고
    • A remark on a Harnack inequality for degenerate parabolic equations
    • F. Chiarenza and R. Serapioni, A remark on a Harnack inequality for degenerate parabolic equations, Rend. Sem. Mat. Univ. Padova 73 (1985), 179-190.
    • (1985) Rend. Sem. Mat. Univ. Padova , vol.73 , pp. 179-190
    • Chiarenza, F.1    Serapioni, R.2
  • 23
    • 0000811231 scopus 로고
    • Intrinsic ultracontractivity of Dirichlet Laplacians in non-smooth domains
    • F. Ciapriani, Intrinsic ultracontractivity of Dirichlet Laplacians in non-smooth domains, Potential Anal. 3 (1994), 203-218.
    • (1994) Potential Anal , vol.3 , pp. 203-218
    • Ciapriani, F.1
  • 24
    • 0042633317 scopus 로고
    • Variétés riemannienes isométriques à l'infini
    • T. Coulhon and L. Salloff-Coste, Variétés riemannienes isométriques à l'infini, Rev. Mat. Iberoamericana 11 (1995), 687-726.
    • (1995) Rev. Mat. Iberoamericana , vol.11 , pp. 687-726
    • Coulhon, T.1    Salloff-Coste, L.2
  • 25
    • 0017727486 scopus 로고
    • Estimates of harmonic measure
    • B. E. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), 275-282.
    • (1977) Arch. Rational Mech. Anal , vol.65 , pp. 275-282
    • Dahlberg, B.E.1
  • 26
    • 44949274310 scopus 로고
    • Intrinsic ultracontractivity and the Dirichlet Laplaciana
    • B. Davis, Intrinsic ultracontractivity and the Dirichlet Laplaciana, J. Funct. Anal. 100 (1991), 162-180.
    • (1991) J. Funct. Anal , vol.100 , pp. 162-180
    • Davis, B.1
  • 28
    • 51249170233 scopus 로고
    • Heat kernel bounds, conservation of probability and the Feller property
    • E. B. Davies, Heat kernel bounds, conservation of probability and the Feller property, J. Analyse Math. 58 (1992), 99-119.
    • (1992) J. Analyse Math , vol.58 , pp. 99-119
    • Davies, E.B.1
  • 29
    • 0001563743 scopus 로고    scopus 로고
    • Non-Gaussian aspects of heat kernel behaviour
    • E. B. Davies, Non-Gaussian aspects of heat kernel behaviour, J. London Math. Soc. 55 (1997), 105-125.
    • (1997) J. London Math. Soc , vol.55 , pp. 105-125
    • Davies, E.B.1
  • 30
    • 0000573290 scopus 로고
    • Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians
    • E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal. 59 (1984), 335-395.
    • (1984) J. Funct. Anal , vol.59 , pp. 335-395
    • Davies, E.B.1    Simon, B.2
  • 31
    • 0001921997 scopus 로고
    • Maximum principle for parabolic inequalities and the heat flow on open manifolds
    • J. Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J. 32 (1983), 703-716.
    • (1983) Indiana Univ. Math. J , vol.32 , pp. 703-716
    • Dodziuk, J.1
  • 32
    • 33646917618 scopus 로고
    • A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash
    • E. B. Fabes and D. W. Stroock, A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal. 96 (1986), 327-338.
    • (1986) Arch. Rational Mech. Anal , vol.96 , pp. 327-338
    • Fabes, E.B.1    Stroock, D.W.2
  • 33
    • 84972539566 scopus 로고
    • On the Martin boundary of Riemannian products
    • A. Freire, On the Martin boundary of Riemannian products, J. Differential Geom. 33 (1991), 215-232.
    • (1991) J. Differential Geom , vol.33 , pp. 215-232
    • Freire, A.1
  • 34
    • 0010786402 scopus 로고
    • The Martin boundary of NTA strips
    • S. J. Gardiner, The Martin boundary of NTA strips, Bull. London Math. Soc. 22 (1990), 163-166.
    • (1990) Bull. London Math. Soc , vol.22 , pp. 163-166
    • Gardiner, S.J.1
  • 35
    • 0003549965 scopus 로고
    • "Elliptic Partial Differential Equations of Second Order"
    • Springer, Berlin
    • D. Gilberg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer, Berlin, 1977.
    • (1977)
    • Gilberg, D.1    Trudinger, N.2
  • 36
    • 0003099262 scopus 로고
    • The Martin compactification of the Cartesian product of two hyperbolic spaces
    • S. Giulini and W. Woes, The Martin compactification of the Cartesian product of two hyperbolic spaces, J. Reine Angew. Math. 444 (1993), 17-28.
    • (1993) J. Reine Angew. Math , vol.444 , pp. 17-28
    • Giulini, S.1    Woes, W.2
  • 37
    • 84956103754 scopus 로고
    • The heat equation on non-compact Riemannian manifolds
    • A. Grigor'yan, The heat equation on non-compact Riemannian manifolds, Math. USSR Sb. 72 (1992), 47-77.
    • (1992) Math. USSR Sb , vol.72 , pp. 47-77
    • Grigor'yan, A.1
  • 38
    • 0000522295 scopus 로고
    • Integral maximum principle and its applications
    • A. Grigor'yan, Integral maximum principle and its applications, Proc. Roy. Soc. Edinburgh 124 A (1994), 353-362.
    • (1994) Proc. Roy. Soc. Edinburgh , vol.124 A , pp. 353-362
    • Grigor'yan, A.1
  • 39
    • 0001436849 scopus 로고
    • Heat kernel upper bounds on a complete non-compact manifold
    • A. Grigor'yan, Heat kernel upper bounds on a complete non-compact manifold, Rev. Mat. Iberoamericana 10 (1994), 395-452.
    • (1994) Rev. Mat. Iberoamericana , vol.10 , pp. 395-452
    • Grigor'yan, A.1
  • 40
    • 0001527051 scopus 로고
    • Heat kernel on a manifold with a local Harnack inequality
    • A. Grigor'yan, Heat kernel on a manifold with a local Harnack inequality, Comm. Anal. Geom. 2 (1994), 111-138.
    • (1994) Comm. Anal. Geom , vol.2 , pp. 111-138
    • Grigor'yan, A.1
  • 41
    • 0039657135 scopus 로고    scopus 로고
    • Analytic and geometric background of recurrence and non-explosion of the Brownian notion on Riemannian manifolds
    • A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian notion on Riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999), 135-249.
    • (1999) Bull. Amer. Math. Soc , vol.36 , pp. 135-249
    • Grigor'yan, A.1
  • 42
    • 0000083908 scopus 로고    scopus 로고
    • Estimates of heat kernels on Riemannian manifolds
    • (B. Davies et al., Eds.), London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge
    • A. Grigor'yan, Estimates of heat kernels on Riemannian manifolds, in "Spectral Theory and Geometry" (B. Davies et al., Eds.), London Mathematical Society Lecture Note Series, Vol. 273, pp. 140-225, Cambridge University Press, Cambridge, 1999.
    • (1999) "Spectral Theory and Geometry" , vol.273 , pp. 140-225
    • Grigor'yan, A.1
  • 43
    • 0032283849 scopus 로고    scopus 로고
    • A Liouville property for Schrödinger operators
    • A. Grigor'yan, and W. Hansen, A Liouville property for Schrödinger operators, Math. Ann. 312 (1998), 659-716.
    • (1998) Math. Ann , vol.312 , pp. 659-716
    • Grigor'yan, A.1    Hansen, W.2
  • 45
    • 0001634126 scopus 로고
    • Mean value and Harnack inequalities for degenerate parabolic equations
    • C. E. Gutiérrez and R. L. Wheeden, Mean value and Harnack inequalities for degenerate parabolic equations, Colloq. Math. LX/LXI (1990), 157-194.
    • (1990) Colloq. Math , vol.60-61 , pp. 157-194
    • Gutiérrez, C.E.1    Wheeden, R.L.2
  • 48
    • 84972562847 scopus 로고
    • Green functions for the Schrödinger equation with short range potentials
    • I. W. Herbst and Z. Zhao, Green functions for the Schrödinger equation with short range potentials, Duke Math. J. 59 (1989), 475-519.
    • (1989) Duke Math. J , vol.59 , pp. 475-519
    • Herbst, I.W.1    Zhao, Z.2
  • 49
    • 84968520667 scopus 로고
    • On the boundary values of harmonic functions
    • R. A. Hunt and R. L. Wheeden, On the boundary values of harmonic functions, Trans. Amer. Math. Soc. 132 (1968), 307-322.
    • (1968) Trans. Amer. Math. Soc , vol.132 , pp. 307-322
    • Hunt, R.A.1    Wheeden, R.L.2
  • 50
    • 84966238197 scopus 로고
    • Positive harmonic functions on Lipschitz domains
    • R. A. Hunt and R. L. Wheeden, Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc. 147 (1970), 505-527.
    • (1970) Trans. Amer. Math. Soc , vol.147 , pp. 505-527
    • Hunt, R.A.1    Wheeden, R.L.2
  • 51
    • 0000738895 scopus 로고    scopus 로고
    • Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains
    • K. Ishige and M. Murata, Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains, Ann. Scuola Norm. Sup. Pisa XXX (2001), 171-223.
    • (2001) Ann. Scuola Norm. Sup. Pisa , vol.30 , pp. 171-223
    • Ishige, K.1    Murata, M.2
  • 52
    • 84966223535 scopus 로고
    • Positive harmonic functions vanishing on the boundary for the Laplacian in unbounded horn-shaped domains
    • D. Ioffe and R. Pinsky, Positive harmonic functions vanishing on the boundary for the Laplacian in unbounded horn-shaped domains, Trans. Amer. Math. Soc. 342 (1994), 773-791.
    • (1994) Trans. Amer. Math. Soc , vol.342 , pp. 773-791
    • Ioffe, D.1    Pinsky, R.2
  • 53
    • 0002155550 scopus 로고    scopus 로고
    • On the behavior of the solutions of degenerate parabolic equations
    • K. Ishige, On the behavior of the solutions of degenerate parabolic equations, Nagoya Math. J. 155 (1999), 1-26.
    • (1999) Nagoya Math. J , vol.155 , pp. 1-26
    • Ishige, K.1
  • 55
    • 0034348848 scopus 로고    scopus 로고
    • Albanese maps and off diagonal long time asymptotics for the heat kernel
    • M. Kotani and T. Sunada, Albanese maps and off diagonal long time asymptotics for the heat kernel, Comm. Math. Phys. 209 (2000), 633-670.
    • (2000) Comm. Math. Phys , vol.209 , pp. 633-670
    • Kotani, M.1    Sunada, T.2
  • 56
    • 0002453196 scopus 로고
    • Large time behavior of the heat equation on complete manifolds with non-negative Ricci curvature
    • P. Li, Large time behavior of the heat equation on complete manifolds with non-negative Ricci curvature, Ann. Math. 124 (1986), 1-21.
    • (1986) Ann. Math , vol.124 , pp. 1-21
    • Li, P.1
  • 57
    • 0000324115 scopus 로고
    • On the parabolic kernel of the Schrödinger operator
    • P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153-201.
    • (1986) Acta Math , vol.156 , pp. 153-201
    • Li, P.1    Yau, S.T.2
  • 58
    • 0002630285 scopus 로고
    • Manifolds with group actions and elliptic operators
    • V. Lin and Y. Pinchover, Manifolds with group actions and elliptic operators, Mem. Amer. Math. Soc. 112 (1994), 1-78.
    • (1994) Mem. Amer. Math. Soc , vol.112 , pp. 1-78
    • Lin, V.1    Pinchover, Y.2
  • 60
    • 84972510206 scopus 로고
    • Harmonic and fullharmonic structures on a differential manifold
    • A-I
    • F.-Y. Maeda, Harmonic and fullharmonic structures on a differential manifold, J. Sci. Hiroshima Univ. Ser. A-I 34 (1970), 271-312.
    • (1970) J. Sci. Hiroshima Univ. Ser , vol.34 , pp. 271-312
    • Maeda, F.-Y.1
  • 61
    • 0000786340 scopus 로고
    • Minimal positive harmonic functions
    • R. S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc. 49 (1941), 137-172.
    • (1941) Trans. Amer. Math. Soc , vol.49 , pp. 137-172
    • Martin, R.S.1
  • 62
    • 0010786629 scopus 로고
    • On the relationship between Martin and Euclidean topologies
    • V. G. Maz'ya, On the relationship between Martin and Euclidean topologies, Soviet Math. Dokl. 18 (1977), 283-286.
    • (1977) Soviet Math. Dokl , vol.18 , pp. 283-286
    • Maz'ya, V.G.1
  • 63
    • 0010787596 scopus 로고    scopus 로고
    • Resolvents and Martin boundaries of product spaces
    • Preprint, Massachusetts Inst. Tech
    • R. Mazzeo and A. Vasy, Resolvents and Martin boundaries of product spaces, Preprint, Massachusetts Inst. Tech., 2000.
    • (2000)
    • Mazzeo, R.1    Vasy, A.2
  • 64
    • 84980078895 scopus 로고
    • A Harnack inequality for parabolic differential equations
    • J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101-134.
    • (1964) Comm. Pure Appl. Math , vol.17 , pp. 101-134
    • Moser, J.1
  • 65
    • 0001361617 scopus 로고
    • Positive solutions and large time behavior of Schrödinger semigroups, Simon's problem
    • M. Murata, Positive solutions and large time behavior of Schrödinger semigroups, Simon's problem, J. Funct. Anal. 56 (1984), 300-310.
    • (1984) J. Funct. Anal , vol.56 , pp. 300-310
    • Murata, M.1
  • 66
    • 84972565480 scopus 로고
    • Large time asymptotics for fundamental solutions of diffusion equations
    • M. Murata, Large time asymptotics for fundamental solutions of diffusion equations, Tôhoku Math. J. 37 (1985), 151-195.
    • (1985) Tôhoku Math. J , vol.37 , pp. 151-195
    • Murata, M.1
  • 68
    • 0001064099 scopus 로고
    • Martin compactification and asymptotics of Green functions for Schrödinger operators with anisotropic potentials
    • M. Murata, Martin compactification and asymptotics of Green functions for Schrödinger operators with anisotropic potentials, Math. Ann. 288 (1990), 211-230.
    • (1990) Math. Ann , vol.288 , pp. 211-230
    • Murata, M.1
  • 69
    • 0001069430 scopus 로고
    • On construction of Martin boundaries for second order elliptic equations
    • M. Murata, On construction of Martin boundaries for second order elliptic equations, Publ. Res. Inst. Math. Sci. Kyoto Univ. 26 (1990), 585-627.
    • (1990) Publ. Res. Inst. Math. Sci. Kyoto Univ , vol.26 , pp. 585-627
    • Murata, M.1
  • 70
    • 0010783009 scopus 로고
    • Positive harmonic functions on rotationary symmetric Riemannian manifolds
    • (M. Kishi, Ed.) Walter de Gruyter, Berlin
    • M. Murata, Positive harmonic functions on rotationary symmetric Riemannian manifolds, in "Potential Theory" (M. Kishi, Ed.), pp. 251-259, Walter de Gruyter, Berlin, 1992.
    • (1992) "Potential Theory" , pp. 251-259
    • Murata, M.1
  • 71
    • 0001220619 scopus 로고
    • Uniform restricted parabolic Harnack inequality, separation principle, and ultracontractivity for parabolic equations
    • "Functional Analysis and Related Topics, 1991" Springer, Berlin
    • M. Murata, Uniform restricted parabolic Harnack inequality, separation principle, and ultracontractivity for parabolic equations, in "Functional Analysis and Related Topics, 1991," Lecture Notes in Mathematics, Vol. 1540, pp. 277-288, Springer, Berlin, 1993.
    • (1993) Lecture Notes in Mathematics , vol.1540 , pp. 277-288
    • Murata, M.1
  • 72
    • 0002164668 scopus 로고
    • Non-uniqueness of the positive Cauchy problem for parabolic equations
    • M. Murata, Non-uniqueness of the positive Cauchy problem for parabolic equations, J. Differential Equations 123 (1995), 343-387.
    • (1995) J. Differential Equations , vol.123 , pp. 343-387
    • Murata, M.1
  • 73
    • 0031498748 scopus 로고    scopus 로고
    • Semismall perturbations in the Martin theory for elliptic equations
    • M. Murata, Semismall perturbations in the Martin theory for elliptic equations, Israel J. Math. 102 (1997), 29-60.
    • (1997) Israel J. Math , vol.102 , pp. 29-60
    • Murata, M.1
  • 74
    • 0010786631 scopus 로고    scopus 로고
    • Structure of positive solutions to Schrödinger equations
    • M. Murata, Structure of positive solutions to Schrödinger equations, Sugaku Expositions 11 (1998), 101-121.
    • (1998) Sugaku Expositions , vol.11 , pp. 101-121
    • Murata, M.1
  • 75
    • 0002351322 scopus 로고    scopus 로고
    • Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds
    • J. R. Norris, Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds, Acta Math. 179 (1997), 79-103.
    • (1997) Acta Math , vol.179 , pp. 79-103
    • Norris, J.R.1
  • 76
    • 0031261349 scopus 로고    scopus 로고
    • Long-time behaviour of heat flow: Global estimates and exact asymptotics
    • J. R. Norris, Long-time behaviour of heat flow: Global estimates and exact asymptotics, Arch. Rational Mech. Anal. 140 (1997), 161-195.
    • (1997) Arch. Rational Mech. Anal , vol.140 , pp. 161-195
    • Norris, J.R.1
  • 77
    • 84957301769 scopus 로고
    • Recurrence criteria for skew products of symmetric Markov processes
    • H. Ôkura, Recurrence criteria for skew products of symmetric Markov processes, Forum Math. 1 (1989), 331-357.
    • (1989) Forum Math , vol.1 , pp. 331-357
    • Ôkura, H.1
  • 78
    • 85031378339 scopus 로고
    • "Lectures on Choquet's theorem"
    • Van Nostrand, Princeton
    • R. Phelps, "Lectures on Choquet's theorem," Mathematical Studies, Vol. 7, Van Nostrand, Princeton, 1965.
    • (1965) Mathematical Studies , vol.7
    • Phelps, R.1
  • 79
    • 0000802818 scopus 로고
    • On criticality and ground states for second order elliptic equations, II
    • Y. Pinchover, On criticality and ground states for second order elliptic equations, II, J. Differential Equation 87 (1990), 353-364.
    • (1990) J. Differential Equation , vol.87 , pp. 353-364
    • Pinchover, Y.1
  • 80
    • 0001247391 scopus 로고
    • Large time behavior of the heat kernel and the behavior of the Green function near criticality for nonsymmetric operators
    • Y. Pinchover, Large time behavior of the heat kernel and the behavior of the Green function near criticality for nonsymmetric operators, J. Funct. Anal. 104 (1992), 54-70.
    • (1992) J. Funct. Anal , vol.104 , pp. 54-70
    • Pinchover, Y.1
  • 81
    • 0001416425 scopus 로고
    • 0-invariant positive harmonic function, a counter example to Stroock's conjecture
    • 0-invariant positive harmonic function, a counter example to Stroock's conjecture, Comm. Partial Differential Equation 20 (1995), 1831-1846.
    • (1995) Comm. Partial Differential Equation , vol.20 , pp. 1831-1846
    • Pinchover, Y.1
  • 82
    • 0033458245 scopus 로고    scopus 로고
    • Maximum and anti-maximum principles and eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations
    • Y. Pinchover, Maximum and anti-maximum principles and eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations, Math. Ann. 314 (1999), 555-590.
    • (1999) Math. Ann , vol.314 , pp. 555-590
    • Pinchover, Y.1
  • 83
    • 0010786409 scopus 로고
    • A new approach to the Martin boundary via diffusions conditioned to hit a compact set
    • R. G. Pinsky, A new approach to the Martin boundary via diffusions conditioned to hit a compact set, Ann. Probab. 21 (1993), 453-481.
    • (1993) Ann. Probab , vol.21 , pp. 453-481
    • Pinsky, R.G.1
  • 86
    • 84972525827 scopus 로고
    • Uniform elliptic operators on Riemmanian manifolds
    • L. Saloff-Coste, Uniform elliptic operators on Riemmanian manifolds, J. Differential Geom. 36 (1992), 417-450.
    • (1992) J. Differential Geom , vol.36 , pp. 417-450
    • Saloff-Coste, L.1
  • 87
    • 84963021360 scopus 로고
    • A note on Poincaré, Sobolev, and Harnack inequality
    • L. Salofff-Coste, A note on Poincaré, Sobolev, and Harnack inequality, Duke Math. J., I.M.R.N. 2 (1992), 27-38.
    • (1992) Duke Math. J., I.M.R.N , vol.2 , pp. 27-38
    • Salofff-Coste, L.1
  • 88
    • 0001586659 scopus 로고
    • Parabolic Harnack inequality for divergence form second order differential operators
    • L. Saloff-Coste, Parabolic Harnack inequality for divergence form second order differential operators, Potential Anal. 4 (1995), 429-467.
    • (1995) Potential Anal , vol.4 , pp. 429-467
    • Saloff-Coste, L.1
  • 89
    • 84966206536 scopus 로고
    • Hölder domains and Poincaré domains
    • W. Smith and D. A. Stegenga, Hölder domains and Poincaré domains, Trans. Amer. Math. Soc. 319 (1990), 67-100.
    • (1990) Trans. Amer. Math. Soc , vol.319 , pp. 67-100
    • Smith, W.1    Stegenga, D.A.2
  • 90
    • 0000195746 scopus 로고
    • Le problème de Dirichlet pour les équationes elliptiques du second ordre à coefficients discontinus
    • G. Stampacchia, Le problème de Dirichlet pour les équationes elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), 189-258.
    • (1965) Ann. Inst. Fourier (Grenoble) , vol.15 , pp. 189-258
    • Stampacchia, G.1
  • 92
    • 84972506714 scopus 로고
    • Analysis on local Dirichlet spaces-II. Upper Gaussian estimates for the fundamental solutions of parabolic equations
    • K.-Th. Sturm, Analysis on local Dirichlet spaces-II. Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math. 32 (1995), 275-312.
    • (1995) Osaka J. Math , vol.32 , pp. 275-312
    • Sturm, K.-Th.1
  • 93
    • 0001581975 scopus 로고    scopus 로고
    • Analysis on local Dirichlet spaces-III. Poincaré and parabolic Harnack inequality
    • K.-Th. Sturm, Analysis on local Dirichlet spaces-III. Poincaré and parabolic Harnack inequality, J. Math. Pures Appl. IX, Ser. 75 (1996), 273-297.
    • (1996) J. Math. Pures Appl. IX, Ser , vol.75 , pp. 273-297
    • Sturm, K.-Th.1
  • 94
    • 84972503485 scopus 로고
    • Related aspects of positivity in Riemannian geometry
    • D. Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom. 25 (1987), 327-351.
    • (1987) J. Differential Geom , vol.25 , pp. 327-351
    • Sullivan, D.1
  • 96
    • 0010920504 scopus 로고    scopus 로고
    • The Martin compactification associated with a second order strictly elliptic partial differential operator on a manifold M
    • (J. C. Taylor, Ed.), CRM Proceedings Lecture Notes American Mathematical Society, Providence, RI
    • J. C. Taylor, The Martin compactification associated with a second order strictly elliptic partial differential operator on a manifold M, in "Topics in Probability and Lie Groups: Boundary Theory" (J. C. Taylor, Ed.), CRM Proceedings Lecture Notes, Vol. 28, pp. 153-202, American Mathematical Society, Providence, RI, 2001.
    • (2001) "Topics in Probability and Lie Groups: Boundary Theory" , vol.28 , pp. 153-202
    • Taylor, J.C.1
  • 97
    • 0001356097 scopus 로고
    • Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains
    • J.-M. G. Wu, Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains, Ann. Inst. Fourier (Grenoble) 28 (1978), 147-167.
    • (1978) Ann. Inst. Fourier (Grenoble) , vol.28 , pp. 147-167
    • Wu, J.-M.G.1
  • 98
    • 0002341788 scopus 로고
    • Subcriticality and gaugeability of the Schrödinger operator
    • Z. Zhao, Subcriticality and gaugeability of the Schrödinger operator, Trans. Amer. Math. Soc. 334 (1992), 75-96.
    • (1992) Trans. Amer. Math. Soc , vol.334 , pp. 75-96
    • Zhao, Z.1


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