On spectral synthesis for contractive \(p\)-norms and Besov spaces (Q1424808)

Let \(F_p\) be a general function space with a contractive \(L_p\)-norm, i.e. we are dealing with norms of the type \[ \int_{(X\times X)\setminus\text{diag}}| u(x)- u(y)|^p\,\nu(dx,dy)+ \int_X| u(x)|^p m(dy). \] For such spaces the following spectral synthesis result is proved: Every quasi-continuous functions in \(F_p\) vanishing quasi-everywhere outside an open set \(G\) can be approximated in this norm by continuous functions in \(F_p\) with compact support in \(G\). This result is applied to contractive Besov spaces of \(d\)-sets in \(\mathbb{R}^N\) and censored stable processes over \(N\)-sets.