An Integration-by-Parts Formula in L1-Spaces

Let d is an element of N and theta is an element of [0, pi/2). Let a(ij) is an element of W-1,W-infinity(R-d, R) for all i, j is an element of {1, ..., d}. Assume C = (a(ij))(1 <= i,j <= d) satisfies (C(x) xi, xi) is an element of Sigma(theta) for all x is an element of R-d and xi is an element of C-d, where Sigma(theta) is the closed sector with vertex 0 and semi-angle theta in the complex plane. Consider the operator A(1) in L-1(R-d) formally given by A(1)u = -Sigma(d)(i,j=1) partial derivative(i)(a(ij) partial derivative(j)u). We prove that A(1) is accretive on W-3,W-1(R-d).