Local Grand Lebesgue Spaces

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.