Boundedness and compactness of convolution operators with unbounded coefficients on locally compact groups
This paper considers the operators arising when convolution operators (with bounded kernels with compact support) and multiplication operators are composed and commuted. The author finds necessary and sufficient conditions that all operators \(A_ fA\), or all operators \(AA_ f\), or all operators \(AA_ f-A_ fA\), etc. be compact, when \(A_ f\) is multiplication by a fixed function f and A varies over the set of all convolution operators with bounded compactly supported kernels. To give the flavour of the results obtained, we mention one interesting application: the Fourier multiplier operator \(A: g\to F^{-1}(a(Fg))\), F denoting the Fourier transformation on \(L^ 2({\mathbb{R}}^ n)\), is of local type, meaning \(AA_ f-A_ fA\) is compact for any continuous f vanishing at \(\infty\), iff a is in \(L^{\infty}({\mathbb{R}}^ n)\) and also \[ \lim_{t\to \infty}\int_{W}| \int_{D}[a(t+x)-a(t+y)] dx|^ 2 dy=0 \] for all relatively compact open sets W and D in \({\mathbb{R}}^ n\).
scientific article; zbMATH DE number 3948964