On the first page of the old paper [Solution of the first boundary value problem for an equation of continuity of an incompressible medium](https://www.researchgate.net/publication/281324404_Solution_of_the_first_boundary_value_problem_for_an_equation_of_continuity_of_an_incompressible_medium) of Bogovskii, the notations $W\_p^l(\Omega)$ and $L\_p^l(\Omega)$ are used without explanation.


I assume that $W\_p^l(\Omega)$ denotes the usual Sobolev space of functions in $L\_p(\Omega) := \{ f : \Omega \to \mathbb{R} : f \textrm{ is measurable and } \int\_{\Omega} |f|^p < \infty \}$ whose weak derivatives of order up to $l$ are also in $L\_p(\Omega)$.


I have not seen the notation $L\_p^l(\Omega)$ used before, but according to the first sentence of this paper it is a "Sobolev space". **What does this notation mean?**



The meaning of $L_p^l(\Omega)$ in a paper of Bogovskii on Sobolev spaces
In [1], chapter 1, §1.1.2 p. 2, $L\_p^l(\Omega)$ is defined as the space of distributions on $\Omega$ whose derivatives of order $l$ belong to the space $L\_p(\Omega)$ as defined above. The relation between $W\_p^l(\Omega)$ and this space is given again in [1], chapter 1, §1.1.4, p. 7 and is
$$
W\_p^l=L\_p^l(\Omega) \cap L\_p(\Omega).
$$


**Reference**


[1] Vladimir G. Maz’ya, *Sobolev spaces. With applications to elliptic partial differential equations*, translated from the Russian by T. O. Shaposhnikova, 2nd revised and augmented edition (English) Grundlehren der Mathematischen Wissenschaften 342. Berlin: Springer (ISBN 978-3-642-15563-5/hbk; 978-3-642-15564-2/ebook), pp. xxviii+866 (2011), [MR2777530](https://mathscinet.ams.org/mathscinet-getitem?mr=MR2777530), [Zbl 1217.46002](https://www.zbmath.org/?q=an%3A1217.46002).


