A cut-off theorem 573

current of bidimension (p,p) in C^, i.e. it is a locally integrable function whose support has Lebesgue 2/?-dimensional measure zero; thus п^{фТ) = 0, i.e.

к . . м . . гЛФ ) = ^Ujf)= ""ЛФТ) (jf) =0.

Thus /i...ki...k = 0; but this holds for every coordinate system therefore (see [Su], р.56)Г=0. D

Let us recall the following

1 . 14 Theorem. Let U be an open subset of IR^ and T be a current on U.

( i ) If T has measure coefficients and dT= 0, then, locally, T= dG for a suitable current G with locally integrable functions as coefficients;

( ii ) If T and dT both have measure coefficients then T is flat;

( iii ) If T is flat and dT = 0, then, locally, T = dGfor a suitable current G with locally integrable functions as coefficients;

( iv ) If T is flat and d-exact on U, then there exists aflat current R on U such that T==dR.

Proof E.g. [Su], pp. 120-121. D

The next theorem corresponds to Theorem 1.14 (i).

1 . 15 Theorem. Let Tbea current ofbidegree (A, k) on Q. If Te JC" {Q) and idUT = 0, then, locally, T= ôG-\-UH for suitable currents G and H with locally integrable functions as coefficients.

Proof Define the operators a and ß as follows

with a := Ö + 3" and ß --= iô^. Now the operator a := (aa*)^ + jS*j? is elliptic ([V], p. 235) and there exists a fundamental solution EofD of the form E = EQ — Qlog\\z\\ where: Eq is a matrix of homogeneous distributions of degree 4 — 2iV, smooth outside 0 e C^, ß is a matrix of polynomials, which is identically zero if 4 < 2iV and constant if 4 = 2iV (see [Ho], Th. 7.1.20). So we get the result simply replacing the Laplace operator with the D operator in the proof of (10.1) in [Su], a

Again the next statement corresponds to Theorem 1.14 (ii) and (iii).

1 . 16 Corollary. Let T be a current ofbidegree (A, k) on Q. (i) If T and idUT both belong to J^* (Й), then T is C-flat.