PARTIAL-DIFFERENTIAL OPERATORS DEPENDING ANALYTICALLY ON A PARAMETER

Let P(lambda,D) = [GRAPHICS] a(alpha)(lambda)D(alpha) be a differential operator with constant coefficients a(alpha) depending analytically on a parameter lambda. Assume that the family {P(lambda,D)} is of constant strength. We investigate the equation P(lambda,D)f(lambda) = g(lambda) where g(lambda) is a given analytic function of lambda with values in some space of distributions and the solution f(lambda) is required to depend analytically on lambda, too. As a special case we obtain a regular fundamental solution of P(lambda,D) which depends analytically on lambda. This result answers a question of L. Hormander.