Multivoice Littlewood-Paley-Meyer wavelets and diagonal dominated pseudodifferential operators (Q1317391)

The paper deals with symbols \(\varepsilon (p)\), \(p \in \mathbb{R}^ d\) of pseudodifferential operators satisfying the condition \(| p^ m D^ m\varepsilon(p) | \leq c\varepsilon(p)\) for some constant \(c\) for all \(p\) and for all multiindices \(m\) with \(| m | \leq d + 1\). For an orthonormal basis \(\psi_ x (p)\) in \(L^ 2\), define the matrix \[ \varepsilon \bigl( x | y \bigr) = (2 \pi)^ -d \int dp \varepsilon (p) \overline {\widehat \psi}_ x(p) \widehat \psi_ y(p). \] The matrix \(\varepsilon (x | y)\) is diagonal dominated if \(\sum_{y \neq x} \varepsilon (x | y) < c \varepsilon (x | x)\) for all \(x\) and some \(c < 1\). The paper gives a construction of a `polygamic' family of wavelets related to Meyer's wavelets (which do not have the desired property), with respect to which the matrix of \(\varepsilon\) is diagonal dominated.