THE HOMOGENEOUS POLYNOMIAL SOLUTIONS FOR THE GRUSHIN OPERATOR

doi:10.1016/S0252-9602(17)30129-7

Abstract

Abstract:

In this paper, the author computes the dimension of space of homogeneous Grushin-harmonic functions, and give an orthogonal basis of them. Moreover, the author describes the nodal curves of these homogenous Grushin-harmonic basis. As an application of the orthogonal basis, the author proves a Liouville-type theorem for the Grushin operator, that is the Grushin-harmonic functions are homogeneous polynomials provided that the frequency of such a function is equal to some constant.

Key words: Grushin operator, frequency function, homogeneous polynomial

Hairong LIU. THE HOMOGENEOUS POLYNOMIAL SOLUTIONS FOR THE GRUSHIN OPERATOR[J].Acta mathematica scientia,Series B, 2018, 38(1): 237-247.

Trendmd

References

[1] D'Ambrosioa L, Lucenteb S. Nonlinear Liouville theorems for Grushin and Tricomi operators. J Differ Equ, 2003, 193(2):511-541
[2] Dunkl C F. An addition theorem for Heisenberg harmonics//Conference on Harmonic Analysis in Honor of Antoni Zygmund. Belmont, CA:Wadsworth International, 1982, 688-705
[3] Franchi B, Gutierrez C E, Wheeden R L. Weighted Sobolev-Poincare inequalities for Grushin type operators. Comm Partial Differ Equ, 1994, 19:523-604
[4] Franchi B, Lanconelli E. Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients. Ann Scuola Norm Sup Pisa, 1983, 4:523-541
[5] Garofalo N. Unique continuation for a class of ellipic operators which degenerate on a manifold of arbitrary codimension. J Differ Equ, 1993, 104(1):117-146
[6] Garofalo N, Shen Z W. Carleman estimates for a subelliptic operator and unique continuation. Ann Inst Fourier, 1994, 44(1):129-166
[7] Garofalo N, Vassilev D. Strong unique continuation properties of generalized Baouendi-Grushin operators. Comm Partial Differ Equ, 2007, 32:643-663
[8] Garofalo N, Rotz K. Properties of a frequency of Almgren type for harmonic functions in Carnot groups. Calc Var Partial Differ Equ, 2015, 54(2):2197-2238
[9] Greiner G, Koornwinder T H. Variations on the Heisenberg spherical harmonics. Stichting Mathematisch Centrum, Zuivere Wiskunde, 1983(ZW 186/83):1-38
[10] Greiner P C. Spherical harmonics on the Heisenberg group. Canad Math Bull, 1980, 23:383-396
[11] Grushin V V. On a class of hypoelliptic operators. Math USSR Sbornik, 1970, 12(3):458-476
[12] Grushin V V. On a class of hypoelliptic pseudodifferential operators degenerate on submanifold. Math USSR Sbornik, 1971, 13(2):155-186
[13] Hörmander L. Hypoelliptic second-order differential equations. Acta Math, 1967, 119:147-171
[14] Lin F H. Nodal sets of solutions of elliptic and parabolic equations. Comm Pure App Math, 1991, XLIV:287-308
[15] Liu H R, Tian L, Yang X P. The growth of H-harmonic functions on the Heisenberg group. Sci China Math, 2014, 57(4):795-806
[16] Liu H R, Tian L, Yang X P. The minimum numbers of nodal domains of spherical Grushin-harmonics. Nonlinear Anal Theory Methods Appl, 2015, 126:143-152
[17] MacRobert T M. Spherical Harmonics, An Elementary Treatise on Harmonic Functions with Applications. Oxford-New York-Toronto:Pergamon Press, 1967
[18] Matsuzawa T. Gevrey hypoellipticity for Grushin operators. Publ Res Inst Math Sci, 1997, 33:775-799
[19] Monti R, Morbidelli D. Kelvin transform for Grushin operators and critical semilinear equations. Duke Math J, 2006, 131(1):167-202
[20] Müller C. Spherical harmonics. Lecture Notes in Mathematics, Vol 17, Berlin, Heidelberg, New York:Springer-Verlag, 1966
[21] Stein E M, Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeion, New Jersey, Princeton University Press, 1971
[22] Song Y Q, Yang X P, Liu Z H. Decomposition of BV functions in Carnot-Carathéodory spaces. Acta Math Sci, 2003, 23B(4):433-439
[23] Song Y Q, Yang X P, Qin J H. Existence of minimisers for a class of free discontinuity problems in the Heisenberg group Hⁿ. Acta Math Sci, 2005, 25B(3):455-469
[24] Sun M B, Yang X P. Generalized Hadamard's inequality and r-convex functions in Carnot groups. J Math Anal Appl, 2004, 294(2):387-398
[25] Sun M B, Yang X P. Inequalities for the weighted mean of r-convex functions. Proc Amer Math Soc, 2005, 133(6):1639-1646
[26] Sun M B, Yang X P. Lipschitz continuity for H-convex functions in Carnot groups. Commun Contemp Math, 2006, 8(1):1-8
[27] Sun M B, Yang X P. Quasi-convex functions in Carnot groups. Chin Ann Math, 2007, 28B(2):235-242
[28] Szegö G. Orthogonal Polynomials. American Mathematical Society, Providence, Rhode Island, 1939
[29] Tian L, Yang X P. Nodal sets and horizontal singular sets of H-harmonic functions on the Heisenberg group. Commun Contemp Math, 2014, 16(4):1-6