Inversion in some algebras of singular integral operators (Q916043)

This paper is devoted to study the algebra \(A_ q\) of Calderón-Zygmund type singular integral operators. Let us denote by \({\mathfrak g}=\oplus_{\alpha \in (0,\infty)}{\mathfrak g}_{\alpha}\), [\({\mathfrak g}_{\alpha},{\mathfrak g}_{\beta}]\subset {\mathfrak g}_{\alpha +\beta}\) a graded finite dimensional nilpotent Lie algebra, \(G=Exp {\mathfrak g}\approx {\mathbb{R}}^ N\) the corresponding connected and simply connected Lie group. For each positive number \(r>0\) one defines a dilation \(\delta_ r\exp \sum X_{\alpha}=\exp (\sum r^{\alpha}X_{\alpha})\) \((X_{\alpha}\in {\mathfrak g}_{\alpha})\). Denote the homogeneous dimension of G by \(d=\sum \alpha \dim {\mathfrak g}_{\alpha}\), as a finitely-supported sum. To any distribution \(\psi\), annihilating constants, on the unit sphere S one can assign a Schwartz class tempered distribution \(K_{\psi}\in PV(G)\subset {\mathcal S}'(G)\) by the singular integral \(<K_{\psi},f>:=\int^{\infty}_{0}<\psi,f_ r>dr/r\in PV(G),\) where \(f_ r(\theta):=f(\delta_ r\theta)\), \(\theta\in S\). This \(K_{\psi}\) is a homogeneous distribution of degree -d. The main result (Theorem 3) answers a question of A. Cabery asserting that the singular integral operators T: \({\mathcal S}(G)\to {\mathcal S}(G)'\), \(T*f:=af+f*K'\), \(a\in {\mathbb{C}}\), \(K'\in PV(G)\), \(K'|_ S\in L^ q(S)\) have the same spectrum with respect to the algebra \(A_ q\) of all such operators or the algebra of all bounded linear operators on \(L^ 2(G)\).