On the semigroups of translations and deformations in anisotropic function spaces with uniform metric (Q2353012)

Let \(\Pi^n=\prod_{m=1}^n(\alpha_m,\beta_m)\subset{\mathbb R}^n\) and let \(\{h_m(x_m)\}_{m=1}^n\) be \(C^1\)-functions on intervals \(\{(\alpha_m,\beta_m)\}_{m=1}^n\) such that \(h'_m>0\) and \(R(h_m)=\mathbb R\). Then the superposition operator \(v(s)= v(h_1(x_1),\dots,h_n(x_n))=u(x_1,\dots,x_n)\), \(v(s)\in C({\mathbb R}^n)\), determines an isometric isomorphism between \(C({\mathbb R}^n)\) and \(C(\Pi^n)\). For \(1<r<n\), define the differential operator \[ L_{r,h}=\sum_{m=1}^r\frac{\partial u}{\partial h_m(x_m)}- \sum_{m=r+1}^n\frac{\partial u}{\partial h_m(x_m)},\quad u\in C(\Pi^n), \] and the one-parameter family of linear operators \[ (U_{r,h}\phi)(x)=\phi(h_1^{-1}[h_1(x_1)+t],\dots,h_r^{-1}[h_r(x_r)+t], h_{r+1}^{-1}[h_{r+1}(x_{r+1})-t],\dots), \] \(\phi(x)\in C(\Pi^n)\). It is proved that \(U_{r,h}\) is a \(C_0\)-semigroup of contractions on \(C(\Pi^n)\) and the operator \(L_{r,h}\) with the domain \[ D(L_{r,h})=\{u\in C(\Pi^n)\mid L_{r,h}u\in C(\Pi^n) \} \] is the generator of the semigroup \(U_{r,h}\).