Function spaces \(W^{1,p}, BV, N^{\lambda,p}\) characterized by the motions of \({\mathbb{R}}^n\) (Q1817937)

The Sobolev spaces \(W^1_p(\mathbb{R}^n)\) with \(1<p\leq\infty\), the space \(\text{BV}(\mathbb{R}^n)\) of functions of bounded variation and the Nikolskij spaces \(N^\lambda_p(\mathbb{R}^n)= B^\lambda_{p,\infty}(\mathbb{R}^n)\) with \(0<\lambda< 1\) and \(1\leq p<\infty\) can be defined via the first differences \(f(x+ h)- f(x)\) where \(x\in\mathbb{R}^n\) and \(h\in\mathbb{R}^n\), that means by the translation group \(\tau_h:x\to x+h\) in \(\mathbb{R}^n\). It is the aim of this paper to replace the translation group by the full group of motions (orientation preserving isometries in \(\mathbb{R}^n\)) in the respective definitions with the outcome of equivalent norms. This is also motivated by some applications in PDE and by replacements of \(\mathbb{R}^n\) by some noncompact homogeneous manifolds.