Abstract
First, the Hardy and Rellich inequalities are defined for the sub-Markovian operator associated with a local Dirichlet form. Secondly, two general conditions are derived which are sufficient to deduce the Rellich inequality from the Hardy inequality. In addition, the Rellich constant is calculated from the Hardy constant. Thirdly, we establish that the criteria for the Rellich inequality are verified for a large class of weighted second-order operators on a domain \(\Omega \subseteq \mathbf{R}^d\). The weighting near the boundary \(\partial \Omega \) can be different from the weighting at infinity. Finally, these results are applied to weighted second-order operators on \(\mathbf{R}^d\backslash \{0\}\) and to a general class of operators of Grushin type.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agmon, S., Lectures on exponential decay of solutions of second-order elliptic equations. Mathematical Notes 29. Princeton University Press, Princeton, 1982.
Ariyoshi, T., and Hino, M., Small-time asymptotic estimates in local Dirichlet spaces. Elec. J. Prob. 10 (2005), 1236–1259.
Andersson, L.-E., On the representation of Dirichlet forms. Ann. Inst. Fourier 25 (1975), 11–25.
Balinsky, A. A., Evans, W. D., and Lewis, R. L., The Analysis and Geometry of Hardy’s Inequality. Universitext. Springer, New York, 2015.
Bouleau, N., and Hirsch, F., Dirichlet forms and analysis on Wiener space, vol. 14 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1991.
Devyver, B., Fraas, M. and Pinchover, Y., Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon, J. Funct. Anal. 266 (2014), 4422–4489.
Driessler, W., and Summers, S. J., On commutators and selfadjointness. Lett. Math. Phys. 7 (1983), 319–326.
Elst, A. F. M. ter, and Robinson, D. W., Uniform subellipticity. J. Operator Theory 62 (2009), 125–149.
Elst, A. F. M. ter, Robinson, D. W., Sikora, A., and Zhu, Y., Dirichlet forms and degenerate elliptic operators. In Koelink, E., Neerven, J. van, Pagter, B. de, and Sweers, G., eds., Partial Differential Equations and Functional Analysis, vol. 168 of Operator Theory: Advances and Applications. Birkhäuser, 2006, 73–95. Philippe Clement Festschrift.
Faris, W. G., Self-adjoint operators. Lect. Notes in Math. 433. Springer-Verlag, Berlin etc., 1975.
Fukushima, M., Oshima, Y., and Takeda, M., Dirichlet forms and symmetric Markov processes, vol. 19 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1994.
Fefferman, C., and Phong, D. H., Subelliptic eigenvalue problems. In Conference on harmonic analysis in honor of Antoni Zygmund, Wadsworth Math. Ser., 590–606. Wadsworth, Belmont, CA, 1983.
Glimm, J., and Jaffe, A., The \(\lambda \phi _2^4\) quantum field theory without cutoffs. IV. Perturbations of the Hamiltonian. J. Math. Phys. 13 (1972), 1568–1584.
Glimm, J., and Jaffe, A., Quantum Physics. A Functional integral point of view. Springer, New York etc., 1981.
Grillo, G., Hardy and Rellich-Type Inequalities for Metrics Defined by Vector Fields. Pot. Anal. 18 (2003), 187–217.
Lehrbäck, J., and Robinson, D. W., Uniqueness of diffusion on domains with rough boundaries. Nonlinear Analysis: Theory, Methods and Applications 131 (2016), 60–80.
Ma, Z. M., and Röckner, M., Introduction to the theory of (non symmetric) Dirichlet Forms. Universitext. Springer, Berlin etc., 1992.
Roth, J.-P., Formule de représentation et troncature des formes de Dirichlet sur \({\bf R}^{m}\). In Séminaire de Théorie du Potentiel de Paris, No. 2, Lect. Notes in Math. 563, 260–274. Springer, Berlin, 1976.
Reed, M., and Simon, B., Methods of modern mathematical physics II. Fourier analysis, self-adjointness. Academic Press, New York etc., 1975.
Robinson, D. W., Hardy inequalities, Rellich inequalities and local Dirichlet forms. (2017) arXiv:1701.05629 math.AP.
Robinson, D. W., and Sikora, A., Analysis of degenerate elliptic operators of Grušin type. Math. Z. 260 (2008), 475–508.
Robinson, D. W., and Sikora, A., Analysis of degenerate elliptic operators of Grušin type. (2008) arXiv:1305.7002 math.AP.
Robinson, D. W., and Sikora, A., Markov uniqueness of degenerate elliptic operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 (2011), 731–759.
Robinson, D. W., and Sikora, A., The limitations of the Poincaré inequality for Grušin type operators. J. Evol. Equ. 14 (2014), 535–563.
Ward, A. D., On Essential Self-adjointness, Confining Potentials and the \(L_p\) -Hardy inequality. PhD thesis, Massey University, Albany, New Zealand, 2014. http://hdl.handle.net/10179/5941.
