Local Grand Lebesgue Spaces
We introduce "local grand" Lebesgue spaces Lp),θx0,a(Ω), 0 < p < ∞, Ω⊆Rn, where the process of "grandization" relates to a single point x0∈Ω, contrast to the case of usual known grand spaces Lp),θ(Ω), where "grandization" relates to all the points of Ω. We define the space Lp),θx0,a(Ω) by means of the weight a(|x-x0|)εp with small exponent, a(0)=0. Under some rather wide assumptions on the choice of the local "grandizer" a(t), we prove some properties of these spaces including their equivalence under different choices of the grandizers a(t) and show that the maximal, singular and Hardy operators preserve such a "single-point grandization" of Lebesgue spaces Lp(Ω), 1 < p < ∞, provided that the lower Matuszewska-Orlicz index of the function a is positive. A Sobolev-type theorem is also proved in local grand spaces under the same condition on the grandizer.