Beals-cordes-type Characterizations of Pseudodifferential Operators

We show that, if U is the representation of SOe(n+1, 1) on L2(Sn) given by (2.11), and P is a bounded operator on L2(Sn), then P belongs to OPS0 1,0(S n) if and only if P (g) = U(g)PU(g)−1 is a C∞ function on SOe(n+1, 1) with values in the Banach space L(L2(Sn)). Introduction Let M be a compact C∞ manifold. Denote by OPS 1,0(M) the space of pseudodifferential operators on M , whose symbols in local coordinates satisfy |Dβ xD ξ p(x, ξ)| ≤ Cαβ〈ξ〉m−|α|, and denote by OPS(M) the subspace of classical pseudodifferential operators, whose symbols have the asymptotic behavior p(x, ξ) ∼ pm(x, ξ) + pm−1(x, ξ) + · · · , |ξ| → ∞, with pm−j(x, ξ) homogeneous of degree m− j in ξ. It was demonstrated in [B], [C], and [D] that, if P : L(M) → L(M), then P belongs to OPS 1,0(M) if and only if, for arbitrary Aj ∈ OPS(M), and any N ∈ Z, adAN · · · adA1 · P : L(M) → L(M), where adAj ·B = [Aj , B]. It is desirable to have some alternative characterizations, and we give one in §2 of this paper, which has as a special case the following (when M = S, the n-dimensional sphere): Theorem. If U is the representation of SOe(n+ 1, 1) on L (S) given by (2.11), and P is a bounded operator on L(S), then P belongs to OPS 1,0(S ) if and only if P (g) = U(g)PU(g)−1 is a C∞ function on SOe(n+ 1, 1) with values in the Banach space L(L2(Sn)). Received by the editors July 5, 1995 and, in revised form, December 6, 1995. 1991 Mathematics Subject Classification. Primary 35S05. This work was partially supported by the National Science Foundation. c ©1997 American Mathematical Society