# Hardy-Littlewood maximal function for Dirac measure on the torus

This question is concerned with an exercise from the book Classical and Multilinear Harmonic Analysis by Muscalu and Schlag (cf. exercise 2.5. on page 38): The task is to find within a multiplicative constant $M \delta_0$. $\delta_0$ denotes the Dirac measure at $0$ and for a complex Borel measure $\mu\in \mathcal{M}(\mathbb{T})$ on the one dimensional torus $\mathbb{T}$(identified with the interval $[0,1]$), the associated maximal function is given by $(M \mu)(\theta)=\sup_{\theta\in I\subset \mathbb{T}} \frac{|\mu|(I)}{|I|}$. The supremum here is taken over all open intervals $I$ and $|I|$ in the denominator denotes the Lebesgue measure of $I$. Moreover, we are asked to show that the weak $L^1$-bound $|\{ \theta \in \mathbb{T} | (M \mu)(\theta) >\lambda \}| \le \frac{3}{\lambda} \left\|\mu\right\|$, where $\left\|\mu\right\|$ denotes the total variation of $\mu$ and $\lambda>0$, can not be improved in general, using this example. First of all I calculated $(M \delta_0)(\theta)=\begin{cases} \frac{1}{\theta}, \quad \theta<\frac12 \\ \frac{1}{1-\theta}, \quad \theta\geq\frac12 \end{cases}$. Is this correct so far? Then I intend to show that, for certain values of $\lambda$, we obtain equality in the aforementioned weak $L^1$-bound. But I don't know how to get the factor $3$ unfortunately.