Mapping properties of one class of quasielliptic operators

The paper is devoted to the theory of quasielliptic operators. We consider scalar and homogeneous quasielliptic operators L(D_x) with lower terms in the whole space ℝⁿ . Our aim is to study mapping properties of these operators in weighted Sobolev spaces. We introduce a special scale of weighted Sobolev spaces W^l _p,q,σ(ℝⁿ). These spaces coincide with known spaces of Sobolev type for some parameters l, q, σ. We investigate mapping properties of the operators L(D_x) in the spaces W_p,q,σ(ℝⁿ). We indicate conditions for unique solvability of quasielliptic equations and systems in these spaces, obtain estimates for solutions and formulate an isomorphism theorem for quasielliptic operators. To prove our results we construct special regularizers for quasielliptic operators.