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
For citation: Samko, S. G. and Umarkhadzhiev, S. M. Local Grand Lebesgue Spaces, Vladikavkaz Math. J., 2021, vol. 23, no. 4, pp. 96-108.
DOI 10.46698/e4624-8934-5248-n
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