Introduction
In this paper we shall examine reflection of singularities of solutions of firstorder equations of the form in a region 9 with boundary given by y=O; say 9 = X a9. Here G = G ( y ) = G ( y , x , 0,) is a smooth one-parameter family of pseudo-differential operators of order one on a 9, G (y) E PS( 1); u takes on values in a vector space, @, and G is a k X k matrix of operators, with principal symbol G,(y,x,t), homogeneous of degree one in 5. On the boundary y = 0, a boundary condition is prescribed :
where P EPS(0) is a pseudo-differential operator of order zero.
We make the assumption that P(y,x,q,t)=det(q-iG,(y,x,t)) is real and has simple characteristics. Then, as is well known (see [ 1 I), singularities of solutions to (1.1) propagate along the null bicharacteristic strips of p in the interior of 9.
Actually, the reference does not quite apply, since a/ay-G is not a pseudodifferential operator on 9 (see the appendix).
Suppose (xo,to) E T*( a s2) -0 and that j null-bicharacteristic strips of p pass over (xo,to). That means there a r e j real solutions q,,. . . ,q, of p ( O , ~~, q , [ ~) = 0 . The associated bicharacteristics y, ( t) = (y (t), x ( t), (t), t( t)) solve the equations *Results obtained at the Courant Institute of Mathematical Sciences, New York University, with the National Science Foundation, Grant NSF-GP-37069X. Reproduction in whole or in part is permitted for any purpose of the United States Government.