operator norm calculation

Question

I have to show that $\Vert{T_{N}(f)}\Vert_{L^p(\mathbb{T})}=1$ $\forall p\in[1,\infty]$ where $T_{N}(f)=\frac{1}{N}\sum_{n=0}^{N-1}f(x+n\alpha)$ with $\alpha\in\mathbb{R}$ and $f$ defined on $\mathbb{T}$ and $\int_{0}^{1}f(x)dx=0$.

I did the cases $p\in({1,2,\infty})$.

How can i extend the result to the other cases? Can I use some interpolation theorem?

Answer 1

I think we can use the interpolation theorem to extend the result to other cases, i.e., $p=1$ and $p=2$.

To see this, note that you have already proved the result for $p=1,2,$ and $\infty$. By interpolation, we have $$ \|T_N(f)\|_{L^p(\mathbb{T})} \leq \|T_N(f)\|_{L^1(\mathbb{T})}^{1-\theta} \|T_N(f)\|_{L^2(\mathbb{T})}^{\theta}, $$ where $\theta \in (0,1)$ and $\frac{1}{p} = \frac{1-\theta}{1} + \frac{\theta}{2}$.

Using the result for $p=1$ and $p=2$, we get \begin{align*} \|T_N(f)\|_{L^p(\mathbb{T})} &\leq 1^{1-\theta} \cdot 1^{\theta/2} = 1, \quad \text{if } p \in [1,2],\\ \|T_N(f)\|_{L^p(\mathbb{T})} &\leq 1^{1-\theta} \cdot 1^{\theta} = 1, \quad \text{if } p \in [2,\infty]. \end{align*}

Therefore, we have $\|T_N(f)\|_{L^p(\mathbb{T})} \leq 1$ for all $p \in [1,\infty]$. On the other hand, since $T_N(f)$ is a normalized function, we also have $\|T_N(f)\|_{L^p(\mathbb{T})} \geq 1$ for all $p \in [1,\infty]$. Hence, we conclude that $\|T_N(f)\|_{L^p(\mathbb{T})} = 1$ for all $p \in [1,\infty]$.

==========UPDATE========= To show that $\|T_N(f)\|_{L^p(\mathbb{T})} = 1$ for all $p \in [1,\infty]$, it suffices to show that $\|T_N(f)\|_{L^p(\mathbb{T})} \leq 1$ and $\|T_N(f)\|_{L^p(\mathbb{T})} \geq 1$ for all $p \in [1,\infty]$.

You have already shown that $\|T_N(f)\|_{L^p(\mathbb{T})} \leq 1$ for $p \in \{1,2,\infty\}$. To show that $\|T_N(f)\|_{L^p(\mathbb{T})} \geq 1$ for all $p \in [1,\infty]$, note that $T_N(f)$ is a normalized function, i.e., $\|T_N(f)\|_{L^1(\mathbb{T})} = 1$. By Holder's inequality, we have $$ \|T_N(f)\|_{L^p(\mathbb{T})} \geq \|T_N(f)\|_{L^1(\mathbb{T})}^{1/p} = 1, $$ for all $p \in [1,\infty]$. Therefore, we have $\|T_N(f)\|_{L^p(\mathbb{T})} = 1$ for all $p \in [1,\infty]$.