A functional calculus for a matrix perturbation of ddx

AbstractWe construct a functional calculus for the operator D(J+A)(i(d/dx)) where J is an invertible n × n constant matrix and A(x) is a small bounded matrix valued function. We define φ(D) and prove its boundedness on L2 for a certain class of φ. Unlike the scalar version of D, it is not clear how to estimate the resolvent of the matrix case in order to apply standard Calderon-Zygmund theory. Instead, using some ideas of Alan McIntosh, we prove boundedness by establishing quadratic estimates. The functional calculus for D leads to a functional calculus for Πi = 1n((− i(ddx)) − ai) where the ai are derivatives of suitably constrained bounded functions. In the last part of the paper, we consider a matrix ∂∂z operator which leads naturally to a bilinear operator the boundedness of which has long been an open question