Removable Sets for Pointwise Solutions of the Generalized Cauchy-Riemann Equations

摘要

The generalized Cauchy-Riemann equations in Euclidean N-space, N > or = 2, in this paper is taken to be the first order differential system: the summation from j = 1 to N of (partial (u sub j))/partial (x sub j) = 0, partial (u sub x(j))/partial (x sub k) - partial (u sub x(k))/partial (x sub j) = 0, j not = k j,k = 1,...N. The notion for a vector field u(x) = (u sub 1(x),...,u sub N(x)) to be a pointwise solution of the generalized Cauchy-Riemann equations is defined in terms of the pointwise L sup 1 total differential of order one. The class A(x sup i, r sub i) is introduced for the ball B(x sup i, r sub i) and the following theorem is established: Let Q be a Borel set of Lebesque measure zero contained in B(x sup i, r sub i). Then a necessary and sufficient condition that Q be removable for the generalized Cauchy-Riemann equations with respect to the class A(x sup i, r sub i) is that Q be countable. It is also shown that the class A(x sup i, r sub i) is in a certain best possible for the sufficienty of the above theorem. (Author)