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 transmission problem which has not hitherto been dealt with in a L_p setting. By educing our problem to one for ordinary differential equations with the aid of the Fourier transformation, we are able to resolve the transmission 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 p ecisely what is required for the furthe development of the spectral theory for the boundary problems cited above.