Hardy-Rellich Type Inequalities Associated with Dunkl Operators

Abstract

In this paper, the authors obtain the Dunkl analogy of classical L ^p Hardy inequality for p > N + 2γ with sharp constant ${\left({{{p - N - 2\gamma} \over p}} \right)^p}$, where 2γ is the degree of weight function associated with Dunkl operators, and L ^p Hardy inequalities with distant function in some G-invariant domains. Moreover they prove two Hardy-Rellich type inequalities for Dunkl operators.

Keywords

Hardy inequalities / Hardy-Rellich inequalities / Best constant / Dunkl operators

References

[1] Ancona A. On strong barriers and an inequality of Hardy for domains in ℝ². J. Lond. Math. Soc., 1986, 34: 274-290

[2] Balinsky A A, Evans W D. Some recent results on Hardy-type inequalities. Appl. Math. Inf. Sci., 2010, 4(2): 191-208

[3] Balinsky A A, Evans W D, Lewis R T. The Analysis and Geometry of Hardy’s Inequality, Universitext, 2015, Cham: Springer-Varleg

[4] Barbatis G, Filippas S, Tertikas A. A unified approach to improved L ^p Hardy inequalities with best constants. Trans. Amer. Math. Soc., 2001, 356: 2169-2196

[5] Brezis H, Marcus M. Hardy’s inequalities revisited. Annali della Scuola normale superiore di Pisa, Classe di scienze, 1997, 25(1): 217-237

[6] Dai F, Xu Y. Analysis on h-Harmonics and Dunkl Transforms, Advanced Courses in Mathematics, CRM Barcelona, 2015, Basel: Birkhauser/Springer-Verlag

[7] D’Ambrosio L, Dipierro S. Hardy inequalities on Riemannian manifolds and applications. Ann. Inst. H. Poinc. Anal. Non Lin., 2014, 31: 449-475

[8] Davies E B. The Hardy constant. Q. J. Math. Oxf., 1995, 46(2): 417-431

[9] Giga Y, Pisante G. On representation of boundary integrals involving the mean curvature for mean-convex domains, 2013, Pisa: Ed. Norm. 171-187

[10] Gromov M. Sign and geometric meaning of curvature. Rend. Sem. Mat. Fis. Milano, 1991, 61(1): 9-123

[11] Hardy G H. An inequality between integrals. Messenger Math., 1925, 54: 150-156

[12] Jerome A, Goldstein I K. The Hardy inequality and nonlinear parabolic equations on Carnot groups. Nonlinear Analysis, 2008, 69: 4643-4653

[13] Jin Y Y. Hardy-type inequalities on the H-type groups and anisotropic Heisenberg groups. Chin. Math. Ann. B, 2008, 29(5): 567-574

[14] Jin Y Y, Han Y Z. Weighted rellich inequality on H-type groups and nonisotropic heisenberg groups. Journal of Inequalities and Applications, 2010, 2010: 1-17

[15] Jin Y Y, Shen S F. Weighted hardy and rellich inequality on carnot groups. Archiv Der Mathematik, 2011, 96: 263-271

[16] Keller M, Pinchover Y, Pogorzelski F. Optimal hardy inequalities for Schrödinger operators on graphs. Communications in Mathematical Physics, 2016, 358(3): 1-24

[17] Lewis R T, Li J, Li Y. A geometric characterization of a sharp Hardy inequality. Journal of Functional Analysis, 2012, 262(7): 3159-3185

[18] Marcus M, Mizel V, Pinchover Y. On the best constant for Hardy’s inequality in ℝ ⁿ. Transactions of the American Mathematical Society, 1998, 350(8): 3237-3255

[19] Maz’ya V G. Sobolev Spaces, 1985, Berlin: Springer-Verlag

[20] Psaradakis G. L¹ Hardy inequalities with weights. J. Geom. Anal., 2013, 23: 1703-1728

[21] Rösler M. Dunkl Operators: Theory and Applications. Lecture Notes in Mathematics, Springer-Verlag, 2002, 1817: 93-135

[22] Tertikas A, Zographopoulos N. Best constant in the Hardy-Rellich inequalities and related improvements. Adv. Math., 2007, 209: 407-459

[23] Velicu, A., Hardy-type inequalities for Dunkl operators, 2019, arXiv:1901.08866v2.