The structure of universal functions for $L^p$-spaces, $p\in(0,1)$

The paper sheds light on the structure of functions which are universal for $L^p$-spaces, $p\in(0,1)$, with respect to the signs of Fourier-Walsh coefficients. It is shown that there exists a measurable set $E\subset [0,1]$, whose measure is arbitrarily close to $1$, such that by an appropriate change of values of any function $f\in L^1[0,1]$ outside $E$ a function $\widetilde f\in L^1[0,1]$ can be obtained that is universal for each $L^p[0,1]$-space, $p\in(0,1)$, with respect to the signs of Fourier-Walsh coefficients.
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