Sharp estimates for Hardy operators on Heisenberg group

Abstract

In the setting of the Heisenberg group, based on the rotation method, we obtain the sharp (p, p) estimate for the Hardy operator. It will be shown that the norm of the Hardy operator on L^p(Hⁿ) is still p/(p−1). This goes some way to imply that the L^p norms of the Hardy operator are the same despite the domains are intervals on ℝ, balls in ℝⁿ, or ‘ellipsoids’ on the Heisenberg group Hⁿ. By constructing a special function, we find the best constant in the weak type (1,1) inequality for the Hardy operator. Using the translation approach, we establish the boundedness for the Hardy operator from H¹ to L¹. Moreover, we describe the difference between M^p weights and A^p weights and obtain the characterizations of such weights using the weighted Hardy inequalities.

Keywords

Heisenberg group / Hardy operator / M_p weight

References

[1] Bényi Á, Oh T. Best constants for certain multilinear integral operators. J Inequal Appl, 2006, Art ID 28582, 12pp

[2] Calderón A P. Inequalities for the maximal function relative to a metric. Studia Math, 1976, 57: 297–306

[3] Carcía-Cuerva J, Rubio de Francia J L. Weighted Norm Inequalities and Related Topics. North Holland Math Studies, Vol 116. Amsterdam: North-Holland Publishing Co, 1985

[4] Chirst M, Grafakos L. Bestconstants for two non-convolution inequalities. Proc Amer Math Soc, 1995, 123: 1687–1693

[5] Coulhon T, Müller D, Zienkiewicz J. About Riesz transforms on the Heisenberg groups. Math Ann, 1996, 305(2): 369–379

[6] Drábek P, Heinig H P, Kufner A. Higher dimensional Hardy inequality. Internat Ser Numer Math, 1997, 123: 3–16

[7] Faris W. Weak Lebesgue spaces and quantum mechanical binding. Duke Math J, 1976, 43: 365–373

[8] Folland G B, Stein E M. Hardy Spaces on Homogeneous Groups. Mathematical Notes, Vol 28. Princeton: Princeton University Press, 1982

[9] Fu Z W, Grafakos L, Lu S Z, Zhao F Y. Sharp bounds for m-linear Hardy and Hilbert operators. Houston J Math, 2012, 38: 225–244

[10] Grafakos L.Montgomery-Smith S. Best constants for uncentred maximal functions. Bull Lond Math Soc, 1997, 29: 60–64

[11] Hardy G H. Note on a theorem of Hilbert. Math Z, 1920, 6: 314–317

[12] Hardy G H, Littlewood J E, Pólya G. Inequalities. Cambridge: Cambridge University Press, 1952

[13] Korányi A, Reimann H M. Quasiconformal mappings on the Heisenberg group. Invent Math, 1985, 80: 309–338

[14] Li H Q. Fonctions maximales centrées de Hardy-Littlewood sur les groupes de Heisenberg. Studia Math, 2009, 191: 89–100

[15] Li H Q, Qian B. Centered Hardy-Littlewood maximal functions on Heisenberg type groups. Trans Amer Math Soc, 2014, 366(3): 1497–1524

[16] Lu S Z, Ding Y, Yan D Y. Singular Integrals and Related Topics. Singapore: World Scientific Publishing Company, 2007

[17] Melas A. The best constant for the centered Hardy-Littlewood maximal inequality. Ann of Math, 2003, 157: 647–488

[18] Muckenhoupt B. Weighted norm inequalities for the Hardy maximal function. Trans Amer Math Soc, 1972, 165: 207–226

[19] Niu P, Zhang H, Wang Y.Hardy type and Rellich type inequalities on the Heisenberg group. Proc Amer Math Soc, 2001, 129: 3623–3630

[20] Stein E M. Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, Vol 43. Princeton: Princeton University Press, 1993

[21] Xiao J. L^p and BMO bounds of weighted Hardy-Littlewood averages. J Math Anal Appl, 2001, 262: 660–666

[22] Zhao F Y, Fu Z W, Lu S Z. M^p weights for bilinear Hardy operators on Rn.Collect Math, 2014, 65: 87–102

[23] Zhao F Y, Fu Z W, Lu S Z. Endpoint estimates for n-dimensional Hardy operators and their commutators. Sci China Math, 2012, 55: 1977–1990

[24] Zienkiewicz J. Estimates for the Hardy-Littlewood maximal function on the Heisenberg group. Colloq Math, 2005, 103: 199–205

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg