Removable Singularities

Abstract

Suppose that $P(x,D) = \sum\nolimits_{\left| \alpha \right| \leqslant p} {{P_\alpha }} {D^\alpha }$ is an (l × k)-matrix of scalar differential operators of order p with C _∞ coefficients on an open set X ⊂ ℝ_n. Every such operator P(x, D) defines a linear mapping P : ε(X) _k → ε(X) _l , locally in the sense that supp Pu ⊂ supp u for all u ∈ ε(X) _k . Via the pairing ε′(X) _k ÷ ε(X) _k → ℂ given by ${(v,u)_X} = \sum\nolimits_{j = 1}^k {{{({v_j},{u_j})}_X}}$ we can identify ε′(X) _k with the topological dual of ε(X) _k . Under this identification, the transpose of the operator P acting on the elements of D(X) _l is defined by a differential operator P′(x, D). This allows one to extend P by the unique way to a continuous linear mapping D′(X) _k → D′(X) _l .