Let \(\Delta\) be the Laplacian and let \(A=(-\Delta)^ m+\rho(x)^{2k}(- \Delta)^{m'}\), where \(m\), \(m'\) and \(k\) are natural numbers, with, say, \(m'>m\), where \(\varrho(x)\) is a bounded function in \(\mathbb{R}^ n\) having there bounded derivatives, and \(\varrho(x)=0\) in some domains is admitted. Let \(1<p<\infty\) and \(j\in \mathbb{N}\), then \[ \| A^ j u\mid L_ p(\mathbb{R}^ n)\|+\| u\mid L_ p(\mathbb{R}^ n)\| \] generates spaces of Sobolev type which are naturally connected with the degenerate elliptic operator \(A\). The paper deals with spaces of this type, their fractional generalizations and Besov type counterparts. All is based on Fourier analytical techniques and pseudodifferential operators (especially of hypoelliptic type of slowly varying strength). The paper gives a thorough study of these spaces, including embedding, interpolation, duality.