Pseudodifferential boundary value problems with non-smooth coefficients

In this contribution, we establish a calculus of pseudodifferential boundary value problems with Holder continuous coefficients. It is a generalization of the calculus of pseudodifferential boundary value problems introduced by Boutet de Monvel. We discuss their mapping properties in Bessel potential and certain Besov spaces. Although having non-smooth coefficients and the operator classes being not closed under composition, we will prove that the composition of Green operators a(1)(x, D-x)a(2)(x ,D-x) coincides with a Green operator a(1)(x , D-x) up to order m(1) + m(2) = theta, where theta is an element of (0, tau(2)) is arbitrary, a(j)( x , xi) is in C-tau j(R-n) w.r.t. x , and m(j) is the order of a(j)(x , D-x), j = 1, 2. Moreover, a(x ,D-x) is obtained by the asymptotic expansion formula of the smooth coefficient case leaving out all terms of order less than m(1) + m(2)-theta. This result is used to construct a parametrix of a uniformly elliptic Green operator a(x ,D-x).