Abstract
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.