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.
Highlights
In this paper we are interested in Hardy, Poincare, Sobolev, Rellich and higher order inequalities of Sobolev–Rellich type in the setting of general homogeneous groups
We define Euler–Hilbert–Sobolev and Sobolev–Lorentz–Zygmund spaces on homogeneous groups
The very convenient framework for working with a given dilation structure is that of homogeneous groups as was initiated in the book [6] of Folland and Stein
Summary
In this paper we are interested in Hardy, Poincare, Sobolev, Rellich and higher order inequalities of Sobolev–Rellich type in the setting of general homogeneous groups. Even in the isotropic situation in Rn, the novelty of the obtained results is in the arbitrariness of the choice of any homogeneous quasinorm, since most of the inequalities are obtained with best constants. In this situation, the very convenient framework for working with a given dilation structure is that of homogeneous groups as was initiated in the book [6] of Folland and Stein. For any complex-valued function f ∈ C0∞(G\{0}) and any α ∈ R we have the following weighted identity:.
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