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 differential equations in partial derivatives. In the case of unbounded domains such spaces are defined with the use of so called grandizers. Under some natural assumptions on the choice of grandizers, we prove the existence, in some grand Sobolev space, of solution to the equation $P_m(D)u(x)=f(x),$ $x\in \mathbb{R}^n,$ $m<n,$ with the right-hand side in the corresponding grand Lebesgue space, where $P_m(D)$ is an elliptic homogeneous differential operator with constant coefficients of even order $m$. Also, for such polynomials 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 lither in terms of spherical hypersingular integrals or in terms of some averages along plane sections of the unit sphere.