I've been self-studying Muscalu and Schlag "Classical and Multilinear Harmonic Analysis" and I would appreciate some help in the following Exercise 5.3.
By explicit expansion and integration for even integers $p$, prove that $$\int_{0}^{1} |\sum_{j=1}^{N} a_{j} r_{j}|^{p}dt \leq C(p) (\sum_{j=1}^{N}|a_{j}^{2}|)^{\frac{p}{2}} $$ where $r_{j}(t)$ here are the Rademacher functions given by $sign(\sin(2\pi 2^{j}t))$ for $t \in (0,1)$ and ${\{a_{j}\}_{j=1}^{N}} \subset \mathbb{C}$. Then, recover the general case of $1 \leq p < \infty$. Show that $C(p) \leq C_{0}\sqrt{p}$ with an absolute constant $C_{0}$ and that this is optimal.
Here's what I tried so far:
Using Newton's multinomial formula $$(\sum_{j=1}^{N} a_{j} r_{j})^{p}dt = \sum_{k_{1}+...+k_{N} = p} {p\choose k_{1},...,k_{N}} a_{1}^{k_{1}}...a_{N}^{k_{N}} r_{1}(t)^{k_{1}}...r_{N}(t)^{k_{N}}$$ Using the symmetry of the Rademacher functions, one can argue for odd $k_{i}$ that $$\mathbb{E}[r_{j}(t)^{k_{i}}]= \int_{0}^{1}r_{j}(t)^{k_{i}} dt = 0$$ and hence using the independence of the Rademacher functions, we have, for $k_{i}$s even only:
$$\mathbb{E}[(\sum_{j=1}^{N} a_{j} r_{j})^{p}] = \sum_{k_{1}+...+k_{N} = p} {p\choose k_{1},...,k_{N}} a_{1}^{k_{1}}...a_{N}^{k_{N}} \mathbb{E}[r_{1}(t)^{k_{1}}]...\mathbb{E}[r_{N}(t)^{k_{N}}] \tag{*}$$
Now, we use the sub-Gaussian bound, proved beforehand in the book: $\mathbb{P}(|\sum\limits_{j=1}^{N}r_{j}a_{j}|>\lambda) \leq 4 exp(-\tfrac{\lambda}{2 \sum\limits_{j=1}^{N} a_{j}^{2}})$ to argue that $\mathbb{E}[r_{1}(t)^{k_{1}}] \leq \mathbb{E}[X_{1}^{k_{1}}] = 2^{\frac{k_{i}}{2}} {k_{i}! \over (\frac{k_{i}}{2})!}$ where $X_{1}$ is a standard normal variable.
Then, we have:
\begin{aligned} \int_{0}^{1} |\sum_{j=1}^{N} a_{j} r_{j}|^{p}dt & \leq \sum_{k_{1}+...+k_{N} = p} {p\choose k_{1},...,k_{N}} a_{1}^{k_{1}}...a_{N}^{k_{N}} \mathbb{E}[X_{1}^{k_{1}}]...\mathbb{E}[X_{N}^{k_{N}}]\\ \end{aligned} for even k_{i}s.
My questions are:
- How to proceed from here to obtain an inequality of the form $ \leq C(p) \sum_{j=1}^{N}|a_{j}^{2}|$? I have the feeling it might be something trivial I am unable to see?
- I'm not sure why I need to assume $p$ is even.
- If the even-ness of p is needed, then how should one proceed for general p. Interpolation theory or Jensen's inequality?
Any hint will also be appreciated to prove the whole question.