### Some Remarks on Harmonic Projection Operators on Spheres

#### Abstract

We give a survey of recent works concerning the mapping properties of joint harmonic projection operators, mapping the space of square integrable functions on complex and quaternionic spheres onto the eigenspaces of the Laplace-Beltrami operator and of a suitably defined subLaplacian. In particular, we discuss similarities and differences between the real, the complex and the quaternionic framework.

#### Keywords

#### Full Text:

PDF (English)#### References

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions: with formulas, graphs, and mathematical tables, Courier Dover Publications, 2012.

F. Astengo, M. G. Cowling and B. Di Blasio, The Cayley transform and uniformly bounded representations, J. Funct. Anal., 213 (2004), 241-269.

F. Baudoin and J.Wang, The Subelliptic Heat Kernels of the Quaternionic Hopf Fibration, Potential Analysis, 41(3) (2014), 959-982.

N. Burq, P. Gerard and N. Tzvetkov, Strichartz inequalities and the non-linear Schrodinger equation on compact manifold, Amer. J. Math. 126 (2004), 569-605.

V. Casarino, Norms of complex harmonic projection operators, Canad. J. Math., 55 (2003), 1134-1154.

V. Casarino, Two-parameter estimates for joint spectral projections on complex spheres, Math. Z., 261 (2009), 245-259.

V. Casarino and P. Ciatti, Transferring Lp eigenfunction bounds from S2n+1 to hn, Studia Math., 194 (2009), 23-42.

V. Casarino and P. Ciatti, Lp joint eigenfunction bounds on quaternionic spheres, to appear in Journal of Fourier Analysis and Applications.

V. Casarino and P. Ciatti, Some harmonic analysis on quaternionic spheres, in preparation.

V. Casarino and M. M. Peloso, Lp - summability of Riesz means for the sublaplacian on complex spheres, J. Lond. Math. Soc. 83 (2011), 137-152.

V. Casarino and M. Peloso, Strichartz estimates and the nonlinear Schrodinger equation for the sublaplacian on complex spheres, Trans. Amer. Math. Soc. 367 (2015), 2631-2664.

D. Geller, The Laplacian and the Kohn Laplacian for the sphere, J. Di. Geom. 15 (1980), 417-435.

D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schrodinger operators (with an appendix by E. M. Stein), Ann. Math., 121 (2) (1985), 463-494.

K. D. Johnson and N. R. Wallach, Composition series and intertwining operators for the spherical principal series. I, Trans. Amer. Math. Soc., 229 (1977), 137-173.

A. U. Klymik and N. Ja. Vilenkin, Representation of Lie groups and special functions. Vol. 2. Class I representations, special functions and integral transforms, Kluwer Academic Publishers, 1993.

B. Kostant, On the existence and the irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627-642.

C. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J., 53 (1986 ), 43-65.

C. Sogge, On the convergence of Riesz means on compact manifolds, Ann. of Math. (2) 126 (1987), 439-447.

C. Sogge, Strong uniqueness theorems for second order elliptic dierential equations, Am. J. Math., 112 (1990), 943-984.

C. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, 105, Cambridge University Press, Cambridge, 1993.

C. Sogge, Lectures on eigenfunctions of the Laplacian, Topics in Mathematical Analysis, Ser. Anal. Appl. Comput., (2008), 337-360.

E. M. Stein, Harmonic Analysis, Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, 1993.

E. M. Stein and G. L. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971.

G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol.23, Amer. Math. Soc., 4th ed. Providence, R.I.(1974).

A.Wunsche, Generalized Zernike or disc polynomials, Journal of Computational and Applied Mathematics, 174 (2005), 135-163.

DOI: 10.6092/issn.2240-2829/6685

### Refbacks

- There are currently no refbacks.

Copyright (c) 2016 Valentina Casarino

This work is licensed under a Creative Commons Attribution 3.0 Unported License.