The meaning of $L_p^l(\Omega)$ in a paper of Bogovskii on Sobolev spaces

Question

On the first page of the old paper 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?

Answer 1

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, Zbl 1217.46002.