Pseudodifferential operators on \(\alpha\)-modulation spaces.
The paper under review studies the behaviour of pseudodifferential opeartors on \(\alpha\)-modulation spaces. The \(\alpha\)-modulation spaces \(M_q^{s, \alpha}(L^p(\mathbb{R}))\), \(1 \leq p,q\leq \infty\), \(s \in \mathbb{R}\), \(0\leq \alpha \leq 1\) are a family of Banach spaces with the norms defined to be the quantity \[ \left( \sum_I (1+| \xi_I| )^{qs} \| {\mathcal F}^{-1} (\psi_I \hat f) \| _p^q \right)^{1/q} < \infty, \] where the family of intervals \(\{I\}\) is a covering of \(\mathbb{R}\) with multiplicity 2 and with the property that for all \(I\) and for all \(\xi \in I\), \(| I| \sim (1+| \xi| )^\alpha\), and where \(\{\psi_I\}\) is an associated partition of unity. For \(q=\infty\), the sum changes to a supremum. It is proven in the paper that certain local cosine bases (called here brushlets, but not identical with the brushlets of Coifman and Meyer) are unconditional bases for \(\alpha\)-modulation spaces. Moreover, a characterization of \(\alpha\)-modulation spaces through expansion coefficients of brushlet bases is provided. This result and the sparsity of brushlet representations are used in the studies of boundedness of pseudodifferential operators on \(\alpha\)-modulation spaces.
scientific article; zbMATH DE number 2104161