The original purpose of this work was to prove the regularity theorems in [1] without using Fourier integral operators. Instead we use the method of Carleman estimates of H6rmander ([3], Ch. 8), with the weight function e ~Cx) replaced by a pseudo-differential operator e r ~ C x ' ~ s~x'~ of variable order s(x, 4). Here ~o(x, 4) and s(x, 4) are homogeneous of degree 0 in ~_. Operators of variable order s(x) were used by Unterberger [10]. These estimates lead to generalizations of various known regularity and existence theorems. For instance we can prove the following theorem.