## Non-harmonic cones are sets of injectivity for the twisted spherical means on $\mathbb {C}^n$

HTML articles powered by AMS MathViewer

- by R. K. Srivastava PDF
- Trans. Amer. Math. Soc.
**368**(2016), 1941-1957 Request permission

## Abstract:

In this article, we prove that a complex cone is a set of injectivity for the twisted spherical means for the class of all continuous functions on $\mathbb C^n$ as long as it does not completely lay on the level surface of any bi-graded homogeneous harmonic polynomial on $\mathbb C^n.$ Further, we produce examples of such level surfaces.## References

- D. H. Armitage,
*Cones on which entire harmonic functions can vanish*, Proc. Roy. Irish Acad. Sect. A**92**(1992), no. 1, 107–110. MR**1173388** - Mark Agranovsky, Carlos Berenstein, and Peter Kuchment,
*Approximation by spherical waves in $L^p$-spaces*, J. Geom. Anal.**6**(1996), no. 3, 365–383 (1997). MR**1471897**, DOI 10.1007/BF02921656 - Gaik Ambartsoumian and Peter Kuchment,
*On the injectivity of the circular Radon transform*, Inverse Problems**21**(2005), no. 2, 473–485. MR**2146272**, DOI 10.1088/0266-5611/21/2/004 - M. L. Agranovsky and E. K. Narayanan,
*$L^p$-integrability, supports of Fourier transforms and uniqueness for convolution equations*, J. Fourier Anal. Appl.**10**(2004), no. 3, 315–324. MR**2066426**, DOI 10.1007/s00041-004-0986-4 - M. L. Agranovsky and E. K. Narayanan,
*A local two radii theorem for the twisted spherical means on $\Bbb C^n$*, Complex analysis and dynamical systems II, Contemp. Math., vol. 382, Amer. Math. Soc., Providence, RI, 2005, pp. 13–27. MR**2175874**, DOI 10.1090/conm/382/07044 - M. L. Agranovskiĭ and E. K. Narayanan,
*Injectivity of the spherical mean operator on the conical manifolds of spheres*, Sibirsk. Mat. Zh.**45**(2004), no. 4, 723–733 (Russian, with Russian summary); English transl., Siberian Math. J.**45**(2004), no. 4, 597–605. MR**2091643**, DOI 10.1023/B:SIMJ.0000035826.91332.06 - Mark L. Agranovsky and Eric Todd Quinto,
*Injectivity sets for the Radon transform over circles and complete systems of radial functions*, J. Funct. Anal.**139**(1996), no. 2, 383–414. MR**1402770**, DOI 10.1006/jfan.1996.0090 - Mark L. Agranovsky and Rama Rawat,
*Injectivity sets for spherical means on the Heisenberg group*, J. Fourier Anal. Appl.**5**(1999), no. 4, 363–372. MR**1700090**, DOI 10.1007/BF01259377 - M. L. Agranovsky, V. V. Volchkov, and L. A. Zalcman,
*Conical uniqueness sets for the spherical Radon transform*, Bull. London Math. Soc.**31**(1999), no. 2, 231–236. MR**1664137**, DOI 10.1112/S0024609398005396 - Charles F. Dunkl,
*Boundary value problems for harmonic functions on the Heisenberg group*, Canad. J. Math.**38**(1986), no. 2, 478–512. MR**833580**, DOI 10.4153/CJM-1986-024-9 - Daryl Geller,
*Spherical harmonics, the Weyl transform and the Fourier transform on the Heisenberg group*, Canad. J. Math.**36**(1984), no. 4, 615–684. MR**756538**, DOI 10.4153/CJM-1984-039-0 - Peter C. Greiner,
*Spherical harmonics on the Heisenberg group*, Canad. Math. Bull.**23**(1980), no. 4, 383–396. MR**602590**, DOI 10.4153/CMB-1980-057-9 - E. K. Narayanan, R. Rawat, and S. K. Ray,
*Approximation by $K$-finite functions in $L^p$ spaces*, Israel J. Math.**161**(2007), 187–207. MR**2350162**, DOI 10.1007/s11856-007-0078-7 - E. K. Narayanan and S. Thangavelu,
*Injectivity sets for spherical means on the Heisenberg group*, J. Math. Anal. Appl.**263**(2001), no. 2, 565–579. MR**1866065**, DOI 10.1006/jmaa.2001.7636 - Vishwambhar Pati and Alladi Sitaram,
*Some questions on integral geometry on Riemannian manifolds*, Sankhyā Ser. A**62**(2000), no. 3, 419–424. Ergodic theory and harmonic analysis (Mumbai, 1999). MR**1803467** - Rama Rawat and A. Sitaram,
*Injectivity sets for spherical means on $\textbf {R}^n$ and on symmetric spaces*, J. Fourier Anal. Appl.**6**(2000), no. 3, 343–348. MR**1755148**, DOI 10.1007/BF02511160 - Walter Rudin,
*Function theory in the unit ball of $\textbf {C}^{n}$*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR**601594** - Rajesh K. Srivastava,
*Sets of injectivity for weighted twisted spherical means and support theorems*, J. Fourier Anal. Appl.**18**(2012), no. 3, 592–608. MR**2921086**, DOI 10.1007/s00041-011-9212-3 - Rajesh K. Srivastava,
*Coxeter system of lines and planes are sets of injectivity for the twisted spherical means*, J. Funct. Anal.**267**(2014), no. 2, 352–383. MR**3210032**, DOI 10.1016/j.jfa.2014.03.009 - Rajesh K. Srivastava,
*Real analytic expansion of spectral projections and extension of the Hecke-Bochner identity*, Israel J. Math.**200**(2014), no. 1, 171–192. MR**3219575**, DOI 10.1007/s11856-014-1041-z - G. Sajith and S. Thangavelu,
*On the injectivity of twisted spherical means on ${\Bbb C}^n$*, Israel J. Math.**122**(2001), 79–92. MR**1826492**, DOI 10.1007/BF02809892 - Elias M. Stein and Guido Weiss,
*Introduction to Fourier analysis on Euclidean spaces*, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR**0304972** - Sundaram Thangavelu,
*An introduction to the uncertainty principle*, Progress in Mathematics, vol. 217, Birkhäuser Boston, Inc., Boston, MA, 2004. Hardy’s theorem on Lie groups; With a foreword by Gerald B. Folland. MR**2008480**, DOI 10.1007/978-0-8176-8164-7 - V. V. Volchkov,
*Injectivity sets for the Radon transform on spheres*, Izv. Ross. Akad. Nauk Ser. Mat.**63**(1999), no. 3, 63–76 (Russian, with Russian summary); English transl., Izv. Math.**63**(1999), no. 3, 481–493. MR**1712132**, DOI 10.1070/im1999v063n03ABEH000248 - V. V. Volchkov,
*Integral geometry and convolution equations*, Kluwer Academic Publishers, Dordrecht, 2003. MR**2016409**, DOI 10.1007/978-94-010-0023-9 - Valery V. Volchkov and Vitaly V. Volchkov,
*Harmonic analysis of mean periodic functions on symmetric spaces and the Heisenberg group*, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2009. MR**2527108**, DOI 10.1007/978-1-84882-533-8 - V. V. Volchkov and Vit. V. Volchkov,
*Functions with vanishing integrals over spheres centered on cones*, Dokl. Akad. Nauk**438**(2011), no. 1, 22–25 (Russian); English transl., Dokl. Math.**83**(2011), no. 3, 298–301. MR**2857371**, DOI 10.1134/S1064562411030094 - Valery V. Volchkov and Vitaly V. Volchkov,
*Offbeat integral geometry on symmetric spaces*, Birkhäuser/Springer Basel AG, Basel, 2013. MR**3024377**, DOI 10.1007/978-3-0348-0572-8 - V. V. Volchkov and Vit. V. Volchkov,
*Behavior at infinity of solutions of a twisted convolution equation*, Izv. Ross. Akad. Nauk Ser. Mat.**76**(2012), no. 1, 85–100 (Russian, with Russian summary); English transl., Izv. Math.**76**(2012), no. 1, 79–93. MR**2951816**, DOI 10.1070/IM2012v076n01ABEH002575

## Additional Information

**R. K. Srivastava**- Affiliation: Department of Mathematics, Indian Institute of Technology, Guwahati, India 781039
- Email: rksri@iitg.ernet.in
- Received by editor(s): December 14, 2013
- Received by editor(s) in revised form: January 11, 2014
- Published electronically: June 18, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**368**(2016), 1941-1957 - MSC (2010): Primary 43A85; Secondary 44A35
- DOI: https://doi.org/10.1090/tran/6488
- MathSciNet review: 3449229