A transmission problem involving a parameter and a weight

In the course of developing a spectral theory for non-selfadjoint elliptic boundary problems involving an indefinite weight function, there arises a t ransmission problem which has not hitherto been dealt with in a L-p setting . By reducing our problem to one for ordinary differential equations with t he aid of the Fourier transformation, we are able to resolve the transmissi on problem, that is to say, we are able to establish L-p estimates for its solutions which are supported in a neighbourhood of the origin, and this is precisely what is required for the further development of the spectral the ory for the boundary problems cited above.