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Injectivity sets for spherical means on the Heisenberg group. (English) Zbl 0995.43003

It is shown that the cylinders \(\Gamma_R=S_R \times\mathbb{R}\), where \(S_R\) is the sphere \(\{z\in\mathbb{C}^n:|z|=R\}\), are sets of injectivity for the spherical means on the Heisenberg group \(\mathbb{H}^n\) in \(L^p\) spaces. On \(\mathbb{R}^n\) the spherical mean of a continuous function \(f\) is defined by \[ f*\sigma_r (x)=\int_{S_r}f(x-y) d\sigma_r(y), \] where \(\sigma_r\) is the normalized surface measure on the sphere \(S_r=\{y\in \mathbb{R}^n: |y|= r\}\). Let \(V\) be a class of functions on \(\mathbb{R}^n\). We say that a subset \(\Gamma\) of \(\mathbb{R}^n\) is a set of injectivity for the spherical mean in \(V\) if \(f*\sigma_r (x)=0\) for all \(r>0\) on \(\Gamma\) implies \(f=0\) for every \(f\in V\). The spheres in \(\mathbb{R}^n\) are not sets of injectivity for the spherical means in \(L^p(\mathbb{R}^n)\) with \(p>{2n \over n-1}\). On the other hand, when \(p\leq {2n\over n-1}\), boundaries of bounded regions in \(\mathbb{R}^n\) are sets of injectivity for the spherical means in \(L^p(\mathbb{R}^n)\). This result has been proved by M. L. Agranovsky, C. Berenstein and P. Kuchment [J. Geom. Anal. 6, 365-383 (1998; Zbl 0898.44003)]. On the Heisenberg group \(\mathbb{H}^n= \mathbb{C}^n \times \mathbb{R}\) the spherical mean of a function \(f\) is defined by \[ f*\mu_R (z,t)= \int_{\mathbb{H}^n} f\Bigl(z-w,t-s- \mathfrak 12\text{Im}(z\cdot \overline w) \Bigr) d\mu_R, \] where \(\mu_R\) is the normalized surface measure on \(S_R \times \{0\}\) in \(\mathbb{H}^n\) and \(S_R=\{z\in\mathbb{C}^n: |z|=R\}\). The authors prove the following theorem:
Theorem A. Let \(1\leq p\leq 2\). Then the cylinder \(\Gamma_R=S_R\times\mathbb{R}\) is a set of injectivity for the spherical means on \(\mathbb{H}^n\) in \(L^p(\mathbb{H}^n)\).
The restriction \(1\leq p\leq 2\) in the theorem is imposed for technical reasons. The authors expect that the above theorem is true for all \(p\), \(1\leq p<\infty\). This theorem is proved from a uniqueness theorem for the heat equation associated to the sub-Laplacian on \(\mathbb{H}^n\). The twisted spherical mean value operator of a function \(f\) on \(\mathbb{C}^n\) is defined by \[ f \times\mu_r (z)=\int_{S_r} f(z-w)e^{(i/2) \text{Im} (z\cdot\overline w)}d \mu_r, \] where \(\mu_r\) is the normalized surface measure on the sphere \(S_r\) on \(\mathbb{C}^n\). For \(1\leq p\leq\infty\), let \(V_p\) be the space of functions satisfying \(f(z)e^{(1/4) |z|^2} \in L^p(\mathbb{C}^n)\). A spherical mean value operator \(M_r\) on \(\mathbb{R}^n\) related to the Hermite operator is defined by \[ M_rf(x)= \int_{S_r}e^{i(x\cdot u+(1/2) u\cdot v)}f (x+v)d\mu_r(w), \] where \(w=u+iv \in \mathbb{C}^n\). For \(1\leq p\leq\infty\), let \(B_p\) be the space of continuous functions on \(\mathbb{R}^n\) for which \(f(x)e^{(1/2) |x|^2} \in L^p( \mathbb{R}^n)\). For these spherical mean value operators the authors prove the following theorem:
Theorem B. Let \(1\leq p\leq\infty\).
(i) The spheres in \(\mathbb{C}^n\) are sets of injectivity for the twisted spherical mean value operator in \(V_p\).
(ii) The spheres in \(\mathbb{R}^n\) are sets of injectivity for the spherical mean value operator \(M_r\) in \(B_p\).
This theorem is proved by using spherical harmonic expansions and a Hecke-Bochner type identity for the Weyl transform which is due to D. Geller [Can. J. Math. 36, 615-684 (1984; Zbl 0596.46034)].
Reviewer: K.Saka (Akita)

MSC:

43A80 Analysis on other specific Lie groups
22E30 Analysis on real and complex Lie groups
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[1] Agranovsky, M. L.; Berenstein, C.; Chang, D.-C.; Pascuas, D., Injectivity of the Pompeiu transform in the Heisenberg group, J. Anal. Math., 63, 131-173 (1994) · Zbl 0808.43002
[2] Agranovsky, M. L.; Berenstein, C.; Kuchment, P., Approximation by spherical waves in \(L^p\) spaces, J. Geom. Anal., 6, 365-383 (1998) · Zbl 0898.44003
[3] Agranovsky, M. L.; Quinto, E. T., Injectivity sets for the Radon transform over circles and complete systems of radial functions, J. Funct. Anal., 139, 383-414 (1996) · Zbl 0860.44002
[4] Agranovsky, M. L.; Rawat, R., Injectivity sets for spherical means on the Heisenberg group, J. Fourier Anal. Appl., 5, 363-372 (1999) · Zbl 0931.43007
[5] Folland, G. B., Harmonic analysis in phase space, Ann. of Math. Stud., 112 (1989) · Zbl 0671.58036
[6] Folland, G. B., Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13, 161-207 (1975) · Zbl 0312.35026
[7] Gaveau, B., Principe de moindre action, propogation de la chaleur, et estimérs sous elliptiques sur certains groupes nilpotents, Acta. Math., 139, 95-153 (1977) · Zbl 0366.22010
[8] Geller, D., Spherical harmonics, the Weyl transform and the Fourier transform on the Heisenberg group, Canad. J. Math., 36, 615-684 (1984) · Zbl 0596.46034
[9] Helgason, S., Geometric Analysis on Symmetric Spaces (1994), Am. Math. Soc: Am. Math. Soc Providence · Zbl 0809.53057
[10] Hulanicki, A., The distribution of energy in the Brownian motion in the Gaussian field and analytic hypoellipticity of certain subelliptic operators on the Heisenberg group, Studia Math., 56, 165-173 (1976) · Zbl 0336.22007
[11] Ratnakumar, P. K.; Thangavelu, S., Spherical means, wave equations and Hermite-Laguerre expansions, J. Funct. Anal., 154, 253-290 (1998) · Zbl 0910.42008
[12] Rawat, R.; Sitaram, A., Injectivity sets for spherical means on \(R^n\) and symmetric spaces, J. Fourier Anal. Appl., 6, 343-348 (2000) · Zbl 0952.43001
[13] Sajith, G.; Thangavelu, S., On the injectivity of twisted spherical means on \(C^n\), Israel J. Math., 122, 79-92 (2001) · Zbl 0986.43001
[14] Szego, G., Orthogonal Polynomials (1967), Am. Math. Soc: Am. Math. Soc Providence · JFM 65.0278.03
[15] Thangavelu, S., Spherical means and CR functions on the Heisenberg group, J. Anal. Math., 63, 255-286 (1994) · Zbl 0822.43001
[16] Thangavelu, S., Lectures on Hermite and Laguerre Expansions (1993), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0791.41030
[17] Thangavelu, S., Harmonic Analysis on the Heisenberg Group. Harmonic Analysis on the Heisenberg Group, Progress in Mathematics, 159 (1998), Birkhäuser: Birkhäuser Boston · Zbl 0892.43001
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