  We define Euler-Hilbert-Sobolev spaces and obtain embedding results on
homogeneous groups using Euler operators, which are homogeneous differential
operators of order zero. Sharp remainder terms of $L^{p}$ and weighted Sobolev
type and Sobolev-Rellich inequalities on homogeneous groups are given. Most
inequalities are obtained with best constants. As consequences, we obtain
analogues of the generalised classical Sobolev type and Sobolev-Rellich
inequalities. We also discuss applications of logarithmic Hardy inequalities to
Sobolev-Lorentz-Zygmund spaces. The obtained results are new already in the
anisotropic $\mathbb R^{n}$ as well as in the isotropic $\mathbb R^{n}$ due to
the freedom in the choice of any homogeneous quasi-norm.
