On the basis property of the Haar system in the space $\mathscr L^{p(t)}([0,1])$ and the principle of localization in the mean

Let $p=p(t)$ be a measurable function defined on $[0,1]$. If $p(t)$ is essentially bounded on $[0,1]$, denote by $\mathscr L^{p(t)}([0,1])$ the set of measurable functions $f$ defined on $[0,1]$ for which $\int_0^1|f(t)|^{p(t)}\,dt<\infty$. The space $\mathscr L^{p(t)}([0,1])$ with $p(t)\geqslant1$ is a normed space with norm
$$ \|f\|_p=\inf\biggl\{\alpha>0:\int\limits_0^1\bigg|\frac{f(t)}\alpha\bigg|^{p(t)}\,dt\leqslant1\biggr\}. $$

This paper examines the question of whether the Haar system is a basis in $\mathscr L^{p(t)}([0,1])$. Conditions that are in a certain sense definitive on the function $p(t)$ in order that the Haar system be a basis of $\mathscr L^{p(t)}([0,1])$ are obtained. The concept of a localization principle in the mean is introduced, and its connection with the space $\mathscr L^{p(t)}([0,1])$ is exhibited.
Bibliography: 2 titles.