Composition Formulas in the Weyl Calculus

Rn×Rn S( x+ y 2 , η) e2iπ〈x−y, η〉 u(y) dy dη : such a linear operator extends as a continuous operator from S ′(Rn) to S(R) while, in the case when S ∈ S ′(Rn ×R) , one can still define Op(S) as a linear operator from S(R) to S ′(Rn) ; also, Op sets up an isometry from L(R×R) onto the space of Hilbert-Schmidt operators on L(R) . The sharp composition S1#S2 of two symbols, say lying in S(R×R) , is that which makes the formula (1.2) Op(S1)Op(S2) = Op(S1#S2) , in which the left-hand side denotes the usual composition of operators, valid.