Construction of function spaces close to $L^{\mathrm{\infty }}$ with associate space close to $L^1$

The paper introduces a variable exponent space $X$ which has in common with $L^{\mathrm{\infty }}([0,1])$ the property that the space $C([0,1])$ of continuous functions on $[0,1]$ is a closed linear subspace in it. The associate space of $X$ contains both the Kolmogorov and the Marcinkiewicz examples of functions in $L^1$ with a.e. divergent Fourier series.