Linear equation with data in non standard spaces

Given a finite family of Banach function spaces V-alpha over a bounded set Omega, V = Pi(alpha), V-alpha, and let T be an element of the dual of the Sobolev space W-2 V. We discuss the existence, uniqueness and regularity of the solution of the linear equation Lu = T under the Dirichlet or Neumann condition on the boundary of Omega. Our results extend recent works on very weak solution with data in weighted distance space or Lorentz space.