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 ∈ ℝⁿ, 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.

中文翻译：

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大空间中的椭圆齐次微分算子

我们将最近几十年来深入研究的所谓大Lebesgue空间和大Sobolev空间应用于偏微分方程。在无界域的情况下，这些空间是使用所谓的“宏化程序”定义的。下上grandizers的选择一些天然假设，我们证明了存在，在一些隆重的Sobolev空间，一个方程的解的P_米（d）û（X）= ˚F（X），X ∈ℝ ^Ñ，M <N，在相应的大Lebesgue空间中位于右侧，其中P _m（D）在一般情况下是任意的椭圆齐次体，我们可以改善一些已知的事实，从而求得算子P _m（D）的基本解：我们可以根据球面的超奇异积分或沿平面的一些平均值以闭合形式构造它单位球面的各个部分。