Sufficiency part For the existence of $\bar p(\cdot)$ $\in$ $W(p)$ for which $C([0,1]$) is closed subspace in $L^{\bar p(\cdot)}([0;1])$.

Question

Handwriting this would be impossible, so I apologize.

These are the definitions and theorems which we need for the proof of the theorem :

BFS is defined as the Bannach Function Space.

Let $W(p)$ denote set of all functions equimeasurable with $p(\cdot)$.

There is the theorem:

My question is all about the sufficiency part of the theorem.

There is the proof of it:

I want to apply this theorem for 2 dimensions.

This will be the statement applied to 2 dimensions:

For the existence of $\bar p(\cdot)$ $\in$ $W(p)$ for which $C([0,1]^2$) is closed subspace in $L^{\bar p(\cdot)}([0;1]^2)$ it is necessary and sufficient that

$\lim_{t\to 0+}$ $ sup \frac {p^{*}(t)}{ln(e/t)}$ $\gt$ $0$.

I've already shown that the theorem $2.2$ and $2.3$ work for two dimensions.

I have a trouble in reconstructing this proof for two dimensions.

I've added an important definition About the BFS and subspaces of BFS X.

Any help would be appreciated.