Lebesgue solvability of equations associated to elliptic and canceling linear differential operators with measure data

Abstract

In this thesis, we present new results on the solvability of the equation A*(x,D) f = µ for f in L^p, with complex measure data µ, associated to an elliptic linear differential operator A(x,D) of order m with variable complex coefficients. Our method is based on (m,p)-energy control of µ giving sufficient conditions for solutions when 1 ≤ p < ∞. A particular study is presented in the global setting of Lebesgue solvability for the equation A*(D) f = µ, where A(D) is a homogeneous differential operator with constant coefficients. We also obtain sufficient conditions in the limiting case p = ∞ using new L^1 (global and local) estimates on measures for elliptic and canceling operators, which are interesting on their own.