This paper deals with the problem of the type triangle open_H f+ f^p =O inquaternionic Heisenberg group, where triangle open_H is the quaternionic Heisenberg Laplacian. Itis proved that, under suitable conditions on p...This paper deals with the problem of the type triangle open_H f+ f^p =O inquaternionic Heisenberg group, where triangle open_H is the quaternionic Heisenberg Laplacian. Itis proved that, under suitable conditions on p and /, the only solution of triangle open_H f+ f^p=O.展开更多
In this paper, we make the asymptotic estimates of the heat kernel for the quaternionic Heisenberg group in various cases. We also use these results to deduce the asymptotic estimates of certain harmonic functions on ...In this paper, we make the asymptotic estimates of the heat kernel for the quaternionic Heisenberg group in various cases. We also use these results to deduce the asymptotic estimates of certain harmonic functions on the quaternionic Heisenberg group. Moreover a Martin compactification of the quaternionic Heisenberg group is constructed, and we prove that the Martin boundary of this group is homeomorphic to the unit ball in the quaternionic field.展开更多
We prove that the restriction operator for the sublaplacian on the quaternion Heisenberg group is bounded from L^p to L^p＇ if 1 ≤ p ≤4/3. This is different from the Heisenberg group, on which the restriction operat...We prove that the restriction operator for the sublaplacian on the quaternion Heisenberg group is bounded from L^p to L^p＇ if 1 ≤ p ≤4/3. This is different from the Heisenberg group, on which the restriction operator is not bounded from Lp to Lp＇ unless p = 1.展开更多
Let be the quaternion Heisenberg group, and let P be the affine automorphism group of . We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of P on...Let be the quaternion Heisenberg group, and let P be the affine automorphism group of . We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of P on L2（ ）. A class of radial wavelets is constructed. The inverse wavelet transform is simplified by using radial wavelets. Then we investigate the Radon transform on . . A Semyanistyi-Lizorkin space is introduced, on which the Radon transform is a bijection. We deal with the Radon transform on both by the Euclidean Fourier transform and the group Fourier transform. These two treatments are essentially equivalent. We also give an inversion formula by using wavelets, which does not require the smoothness of functions if the wavelet is smooth. In addition, we obtain an inversion formula of the Radon transform associated with the sub-Laplacian on .展开更多
文摘This paper deals with the problem of the type triangle open_H f+ f^p =O inquaternionic Heisenberg group, where triangle open_H is the quaternionic Heisenberg Laplacian. Itis proved that, under suitable conditions on p and /, the only solution of triangle open_H f+ f^p=O.
基金the National Natural Science Foundation of China Grant 10261002
文摘In this paper, we make the asymptotic estimates of the heat kernel for the quaternionic Heisenberg group in various cases. We also use these results to deduce the asymptotic estimates of certain harmonic functions on the quaternionic Heisenberg group. Moreover a Martin compactification of the quaternionic Heisenberg group is constructed, and we prove that the Martin boundary of this group is homeomorphic to the unit ball in the quaternionic field.
基金Supported by National Natural Science Foundation of China (10871003, 10990012)the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007001040)
文摘We prove that the restriction operator for the sublaplacian on the quaternion Heisenberg group is bounded from L^p to L^p＇ if 1 ≤ p ≤4/3. This is different from the Heisenberg group, on which the restriction operator is not bounded from Lp to Lp＇ unless p = 1.
基金supported by National Natural Science Foundation of China(Grant Nos.10971039 and 11271091)the second author is supported by National Natural Science Foundation of China(Grant No.10990012)the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.2012000110059)
文摘Let be the quaternion Heisenberg group, and let P be the affine automorphism group of . We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of P on L2（ ）. A class of radial wavelets is constructed. The inverse wavelet transform is simplified by using radial wavelets. Then we investigate the Radon transform on . . A Semyanistyi-Lizorkin space is introduced, on which the Radon transform is a bijection. We deal with the Radon transform on both by the Euclidean Fourier transform and the group Fourier transform. These two treatments are essentially equivalent. We also give an inversion formula by using wavelets, which does not require the smoothness of functions if the wavelet is smooth. In addition, we obtain an inversion formula of the Radon transform associated with the sub-Laplacian on .
