Local grand Lebesgue spaces

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.