## 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.

### MSC:

 26D10 Inequalities involving derivatives and differential and integral operators 60J40 Right processes
Full Text:

### References:

 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.