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DOI: 10.46698/e4624-8934-5248-n

# Local Grand Lebesgue Spaces

Samko, S. G. , Umarkhadzhiev, S. M.
Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 4.
Abstract:
We introduce "local grand" Lebesgue spaces $$L^{p),\theta}_{x_0,a}(\Omega)$$, $$0 < p < \infty,$$ $$\Omega \subseteq \mathbb{R}^n$$, where  the process of "grandization" relates to a single point $$x_0\in \Omega$$, contrast to the case of usual known grand spaces $$L^{p),\theta}(\Omega)$$, where "grandization" relates to all the points of $$\Omega$$.
We define the space $$L^{p),\theta}_{x_0,a}(\Omega)$$ by means of the weight $$a(|x-x_0|)^{\varepsilon p}$$ 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 $$L^p(\Omega)$$, $$1<p<\infty$$, 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.
Keywords: grand space, Lebesgue space, Muckenhoupt weight, maximal operator, singular operator, Hardy operator, Stein-Weiss interpolation theorem, Matuszewska-Orlicz indices