On local solvability of higher order elliptic equations in rearrangement invariant spaces

We consider a higher order elliptic equation with nonsmooth coefficients with respect to rearrangement invariant spaces on the domain $\Omega \subset {\Bbb R}^{n} $. Separable subspaces of these spaces are distinguished in which infinitely differentiable and compactly supported functions are dense; Sobolev spaces generated by these subspaces are determined. Under certain conditions on the coefficients of the equation and the Boyd indices of the rearrangement invariant space, we prove the local solvability of the equation in rearrangement invariant Sobolev spaces. This result strengthens the previously known classical $L_{p} $-analog. Rearrangement invariant spaces include Lebesgue, Marcinkiewicz, grand Lebesgue, Orlicz, Lorentz spaces and many others. We present some results concerning these particular cases and a result related to the weak-$L_{p}^{w} $ space.