This article provides a revised version of some existing results in the literature for the quaternion Fourier transform (QFT) and quaternion wavelet transforms. The inner-product relation and its consequent formula for the continuous quaternion wavelet transform (CQWT) are derived in \(L^p ({\mathbb {R}}^{2}; {\mathbb {H}})\) space under the assumption that the admissible wavelet is complex-valued and has a real QFT. Furthermore, the characterization of quaternion Sobolev spaces \(H^{s}({\mathbb {R}}^{2}; {\mathbb {H}})\) and \(W^{m,p} (\Omega ; {\mathbb {H}})\) , weighted quaternion Sobolev space \(W_{k}^{m,p} (\varvec{\Omega }; {\mathbb {H}} )\) and generalized quaternion Sobolev space \(H_{w}^{\omega } ({\mathbb {R}}^{2}; {\mathbb {H}})\) , quaternion Besov space by means of the CQWT is presented. The CQWT is analysed within these function and distribution spaces, yielding novel findings regarding continuity and boundedness.
