Let （X,d,μ） be a metric measure space satisfying both the upper doubling and the geometrically doubling conditions in the sense of HytSnen. Under this assumption, we prove that θ-type Calderon-Zygmund operators which are bounded on L2（μ） are also bounded from L∞（μ） into RBMO（μ） and from H1,∞at（μ） into L1（μ）.
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 .
