Hardy’s inequality for Dirichlet forms. (English) Zbl 0979.26007

The aim of the paper is to give necessary and sufficient conditions for the validity of an abstract form of Hardy’s \(L^2\) inequality in which the Dirichlet integral is replaced by the Dirichlet form of a general symmetric Markov process. The case of equality in a given inequality is described. The general result is illustrated with examples.


26D10 Inequalities involving derivatives and differential and integral operators
60J40 Right processes
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[1] Ancona, A., On strong barriers and an inequality of Hardy for domains in \(R\)^{n}, J. London math. soc. (2), 34, 274-290, (1986) · Zbl 0629.31002
[2] Beesack, P.R., Hardy’s inequality and its extensions, Pacific J. math., 11, 39-61, (1961) · Zbl 0103.03503
[3] Carron, G., Inégalités de Hardy sur LES variétés riemanniennes non-compactes, J. math. pures appl., 76, 883-891, (1997)
[4] Chernoff, H., A note on an inequality involving the normal distribution, Ann. probab., 9, 533-535, (1981) · Zbl 0457.60014
[5] Fitzsimmons, P.J.; Getoor, R.K., On the potential theory of symmetric Markov processes, Math. ann., 281, 495-512, (1988) · Zbl 0627.60067
[6] Fitzsimmons, P.J., Time changes of symmetric Markov processes and a feynman – kac formula, J. theoret. probab., 2, 487-501, (1989) · Zbl 0683.60052
[7] Fitzsimmons, P.J., Even and odd continuous additive functionals, Dirichlet forms and stochastic processes, Beijing, 1993, (1995), de Gruyter Berlin, p. 139-154 · Zbl 0844.60048
[8] P. J. Fitzsimmons, On the quasi-regularity of semi-Dirichlet forms, Potential Anal, in press. · Zbl 0993.60075
[9] Fukushima, M.; Ōshima, Y.; Takeda, M., Dirichlet forms and symmetric Markov processes, (1994), de Gruyter Berlin · Zbl 0838.31001
[10] Getoor, R.K.; Glover, J., Riesz decompositions in Markov process theory, Trans. amer. math. soc., 285, 107-132, (1984) · Zbl 0547.60076
[11] Getoor, R.K., Markov processes: ray processes and right processes, Lecture notes in mathematics, (1975), Springer-Verlag Berlin/New York · Zbl 0299.60051
[12] Glover, J.; Rao, M.; S̆ikić, H.; Song, R., Quadratic forms corresponding to the generalized Schrödinger semigroups, J. funct. anal., 125, 358-378, (1994) · Zbl 0807.60056
[13] Hansson, K., Imbedding theorems of Sobolev type in potential theory, Math. scand., 45, 77-102, (1979) · Zbl 0437.31009
[14] Hardy, G.H., Notes on some points in the integral calculus LI, Messenger math., 44, 107-112, (1919) · Zbl 0002.13004
[15] Hardy, G.H.; Littlewood, J.E.; Pólya, G., Inequalities, (1952), Cambridge Univ. Press Cambridge
[16] Kato, T., Perturbation theory for linear operators, (1980), Springer-Verlag Berlin/New York
[17] Ma, Z.-M.; Röckner, M., Introduction to the theory of (non-symmetric) Dirichlet forms, Universitext, (1992), Springer-Verlag Berlin
[18] Maz’ja, V.G., Sobolev spaces, (1985), Springer-Verlag Berlin/Heidelberg/New York/Tokyo
[19] Opic, B.; Kufner, A., Hardy-type inequalities, Pitman research notes in mathematics, (1990), Longman Essex
[20] Rao, M.; S̆ikić, H., Potential inequality, Israel J. math., 83, 97-127, (1993) · Zbl 0786.31006
[21] Sharpe, M., General theory of Markov processes, (1988), Academic Press San Diego · Zbl 0649.60079
[22] Stollmann, P.; Voigt, J., Perturbation of Dirichlet forms by measures, Potential anal., 5, 109-138, (1996) · Zbl 0861.31004
[23] Tomaselli, G., A class of inequalities, Boll. un. mat. ital., 2, 622-631, (1969) · Zbl 0188.12103
[24] Vondrac̆ek, Z., An estimate for the L2-norm of a quasi continuous function with respect to a smooth measure, Arch. math., 67, 408-414, (1996) · Zbl 0878.60048
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