We define Euler–Hilbert–Sobolev spaces and obtain embed-
ding 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 dis-
cuss applications of logarithmic Hardy inequalities to Sobolev–Lorentz–
Zygmund spaces. The obtained results are new already in the anisotropic
R
n
as well as in the isotropic
R
n
due to the freedom in the choice of any
homogeneous quasi-norm.