| This work is part of general program for studying the analytic dependence of solutions of partial differential equations on the boundary. Such dependence is usually reduced to an a priori estimate for a non-linear operator. We study the Cauchy Integral Operator, for which we extend the results of A. P. Calderon, R. R. Coifman, Y. Meyer and G. David to obtain weighted L('p) estimates:; (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI); We obtain similar results for the double layer potential singular integral operators in R('n). We also study another possible extension of the one-dimensional results to higher dimensions involving the tensor products of the one-dimensional multilinear operators that were used to study the Cauchy Integral. These can be viewed as Gateaux differentials at 0 of a generalized Cauchy Integral. We give a direct proof of the L('2) boundedness of the operator B(A,F) = {lcub}(VBAR)D(,1)(VBAR), {lcub}(VBAR)D(,2)(VBAR), M(,A){rcub}{rcub}F, which is the tensor product with itself of the Calderon Commutator {lcub}(VBAR)D(VBAR), M(,a){rcub}f. In addition, we provide a counter-example showing that tensor products in this sense of bounded operators are not necessarily bounded. |
