Nuclear function spaces on the base of a hypercomplex system
Let Q be a complete separable locally compact metric space. The authors consider socalled structure measures \(\gamma\) (A,B,r) on Q, where \(r\in Q\) and A,\(B\subseteq Q\) are Borel sets. Satisfying certain properties these measures constitute so called hypercomplex systems with basis Q, which are the spaces \(L_ 1(Q)\) with a convolution defined by \(\gamma\). Let \(\Xi\) be a fixed sequence of functions \(\xi_ n\) in \(L_ 2(Q)\) and \(D_ k=(T_{\xi_ 1}...T_{\xi_ k})^{-1}\) be ''differential'''' operators, where \(T_{\xi_ j}f=\xi_ j*f.\) These operators \(D_ k\) are defined on the range of \(T_{\xi_ 1}...T_{\xi_ k}\). Now let \({\mathcal C}^{\infty}\!_ 0(Q,\Xi)\) denote the space of all infinitely ''differentiable'' functions on Q, where differentiability is to be interpreted as above. For a certain index set \(T=\{\tau =(\tau_ 1,\tau_ 2(p)\}\) denote \(D(Q,\Xi)=\cap_{\tau \in T}H_{\tau},\) where \(H_{\tau}\) is the completion of the space \({\mathcal C}^{\infty}\!_ 0(Q,\Xi)\) with respect to the inner product \[ (f,g)_{\tau}=\sum^{\tau_ 1}_{k=0}\int(D_ kf)(p)(D_ kg)(p)\tau_ 2(p)dp,\quad \tau_ 2:Q\to [1,\infty],\quad \tau_ 1=0,1,...\quad. \] It is shown that under suitable assumptions the space D(Q,\(\Xi)\) is a nuclear space and that D(Q,\(\Xi)\) is invariant under \(T_{\xi}\).
scientific article; zbMATH DE number 3846216
scientific article