We consider the ∂-Dirac system that Ablowitz and Fokas used to transform the defocussing Davey–Stewartson system to a linear evolution equation. The nonlinear Plancherel identity for the scattering transform was established by Beals–Coifman for Schwartz functions. Sung extended the validity of the identity to functions belonging to L(R) ∩ L∞(R2) and Brown to L(R)–functions with sufficiently sma...

We examine the structure of the Levi component MA in a minimal parabolic subgroup P = MAN of a real reductive Lie group G and work out the cases where M is metabelian, equivalently where p is solvable. When G is a linear group we verify that p is solvable if and only if M is commutative. In the general case M is abelian modulo the center ZG , we indicate the exact structure of M and P , and we ...

Given a unimodular real spherical space $Z=G/H$ we construct for each boundary degeneration $Z_I=G/H_I$ of $Z$ Bernstein morphism $B_I: L^2(Z_I)_{\rm disc }\to L^2(Z)$. We show that $B:=\bigoplus_I B_I$ provides an isospectral $G$-equivariant onto $L^2(Z)$. Further, the maps $B_I$ are finite linear combinations orthogonal projections which translates in known cases where is group or symmetric i...