On Elliptic Homogeneous Differential Operators in Grand Spaces

We give an application of so-called grand Lebesgue and grand Sobolev spaces, intensively studied during last decades, to partial differential equations. In the case of unbounded domains such spaces are defined using so-called grandizers. Under some natural assumptions on the choice of grandizers, we prove the existence, in some grand Sobolev space, of a solution to the equation P m ( D ) u ( x ) = f ( x ), x ∈ ℝ n , m < n , with the right-hand side in the corresponding grand Lebesgue space, where P m ( D ) is an arbitrary elliptic homogeneous in the general case we improve some known facts for the fundamental solution of the operator P m ( D ): we construct it in the closed form either in terms of spherical hypersingular integrals or in terms of some averages along plane sections of the unit sphere.