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With Khinchine's inequality, prove Fourier basis is unconditional in $L^{p}[0,1]$ only for $p=2$

MathOverflow Asked by Eric Yan on December 3, 2020

I am trying to prove Problem 6.10 in "Classical and Multilinear Harmonic Analysis" by by Camil Muscalu and Wilhelm Schlag.

Problem

Problem 6.10. Let $1le p < infty$ and suppose that there exists a constant $C(p)$ such that
$$sup_{varepsilon_n=pm1} leftlVert sum_{-N}^N varepsilon_n widehat f(n) e(ntheta) rightrVert le C(p) |f|_p qquad forall fin L^p([0,1]), forall Nge1. tag{6.33}$$
Show that necessarily $p=2$. Next show that (6.33) is equivalent to the property that $sum_{-infty}^infty varepsilon_n widehat f(n) e(ncdot)$ converges in $L^p([0,1])$ for each choice of signs $varepsilon_n=pm1$ and $fin L^p([0,1])$. This latter property is called unconditional convergence, and this problem therefore amounts to proving that the exponential system is unconditional only for $p=2$. In Section 8.4 we shall see that Haar functions are unconditional on $L^p([0,1])$ for $1<p<infty$. Hint: Use Khinchine’s inequality

Here are some attempts,
begin{align*}
mathbb{E}Big|sum_{n=-N}^Nepsilon_nhat f(n)e(ntheta)Big|_p&=mathbb{E}Big(int_0^1|sum_{n=-N}^Nepsilon_nhat f(n)e(ntheta)|^pdthetaBig)^{1/p}\
&leqBig(mathbb{E}int_0^1|sum_{n=-N}^Nepsilon_nhat f(n)e(ntheta)|^pdthetaBig)^{1/p}quad(text{since $g(x)=x^{1/p}$ is concave for $pgeq1$})\
&leq C(sum_{n=-N}^N|hat f(n)|^2)^{1/2}quad(text{Khinchine’s inequality})\
&leq C|f|_2
end{align*}

But
$$
mathbb{E}Big|sum_{n=-N}^Nepsilon_nhat f(n)e(ntheta)Big|_pleqsup_{epsilon_n}Big|sum_{n=-N}^Nepsilon_nhat f(n)e(ntheta)Big|_p
$$

the inequality seems in the wrong direction. Any suggestion on how to do next? Thanks in advance!

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