Invariant subspaces in some function spaces on the light cone in \( \mathbb R^3\). (English. Russian original) Zbl 1275.43007
Sb. Math. 203, No. 6, 864-892 (2012); translation from Mat. Sb. 203, No. 6, 101-130 (2012).
Summary: For certain topological vector spaces of functions on the light cone \(X\) in \(\mathbb R^3\), we obtain a complete description of all the closed linear subspaces which are invariant with respect to the natural quasiregular representation of the group \(\mathbb R\oplus\mathrm{SO}_0(1,2)\). In particular, we give a description of irreducible and indecomposable invariant subspaces. Among the function spaces we consider we include, in particular, the spaces \(C(X)\) and \(\mathcal E(X)\) of continuous and infinitely differentiable functions on \(X\) and also function spaces formed by functions with exponential growth on \(X\).
MSC:
43A45 | Spectral synthesis on groups, semigroups, etc. |
22E30 | Analysis on real and complex Lie groups |
43A85 | Harmonic analysis on homogeneous spaces |