On sharp bounds of some dyadic operators I am now having interest in finding the sharp bounds of some kinds of dyadic operators which map $L^p(\mathbb{R})$ to $L^p(\mathbb{R})$. 
For example, the martingale transform $T_{\sigma}$ which defined as
$$
T_{\sigma}f(x)=\sum_{I\in \mathcal{D}} \sigma_{I}\left< f,h_I \right> h_I(x),
$$
where $\sigma_I = \pm1$, $\mathcal{D}$ is the standard dyadic grid on $\mathbb{R}$ and $h_I(x)$ is the Haar function. It is well known that 
$$
\underset{\sigma}{\sup}\| T_{\sigma}f \|_p \leq C_p \| f \|_p.
$$
Indeed, it has been proved that the best constant $C_p=p^* - 1$ with $p^*=\max\{ p-1, 1/(p-1) \}$.
Analogously, we can define the dyadic shift operator $Ш^{r,\beta}$ as
$$
Ш^{r,\beta}f(x)=\underset{I\in r\mathcal{D}^{\beta}}{\sum} \left< f, h_I \right>H_I(x),
$$
where $r\mathcal{D}^{\beta}$ is the random dyadic grid, and $H_I(x) =1/\sqrt{2}\left( h_{I_{+}}(x) - h_{I_{-}}(x) \right)$. Then we have
$$
\underset{r,\beta}{\sup}\| Ш^{r,\beta}f \|_p \leq C_p \| f \|_p,  ~~~~ 1<p<\infty.
$$
However, I don't know what the best constant $C_p$ is, namely, what is the smallest constant $C_p$ in the above inequality?
Simultaneously, for the dyadic square function defined by
$$
S^{r,\beta}(f)(x)=\left( \underset{I \in r\mathcal{D}^{\beta}}{\sum} \frac{\left| \left< f,h_I \right> \right|^2}{|I|} \chi_{I}(x) \right)^{1/2}.
$$
For $1<p<\infty$, how can we prove the following inequality 
$$
\underset{r,\beta}{\sup}\| S^{r,\beta}f \|_p \leq C_p \| f \|_p ~~? 
$$
In addition, what's the smallest constant $C_p$ for the uniform $L^p$ bound of $S^{r,\beta}$ ? 
Supplement: It is easy to verify that $\| S^{r,\beta}f \|_2 = \| f \|_2$. For the case of $p \ne 2$, it seems that a solution to this sharp estimation would become difficult. 
 A: Let $f \in L^p(\mathbb{R})$ arbitrarily. We know that for $j \in \mathbb{Z}$ fixed there is an unique $I \in r\mathcal{D}^{\beta}_j$ such that $x \in I$ for some $x \in \mathbb{R}$. Define $Q_j[f](x) = \sum_{x \in I \in r\mathcal{D}^{\beta}_{j}} \langle f, h_I \rangle h_I(x)$. So
$$|Q_j[f](x)|^2 = \left|\sum_{x \in I \in r\mathcal{D}^{\beta}_j}\langle f, h_I \rangle h_I(x)\right|^2 = \sum_{x \in I \in r\mathcal{D}^{\beta}_j}\left|\langle f, h_I \rangle h_I(x) \right|^2$$
Then we have that
$$S^{r, \beta}(f)(x) = \left(\sum_{j \in \mathbb{Z}}\left|Q_j[f](x)\right|^2\right)^{1/2}$$
First define a truncation of $S^{r, \beta}(f)$
$$S^{r, \beta}_m(f) := \left(\sum_{|j| \leq m}\left|Q_j[f](x)\right|^2\right)^{1/2}$$
Using Khintchine's Inequality
\begin{align*}
    \lVert S^{r, \beta}_m(f) \rVert_{L^p(\mathbb{R})} &\lesssim_{p} \left\lVert\mathbb{E}\left(\left|\sum_{|j| \leq m}Q_j[f]X_k\right|^p\right)^{1/p}\right\rVert_{L^p(\mathbb{R})} \\
    &= \mathbb{E}\left(\left\lVert\sum_{|j| \leq m} Q_j[f]X_k\right\rVert^p_{L^p(\mathbb{R})}\right)^{1/p} \\
    &\leq \mathbb{E}\left(\left\lVert\sum_{|j| \leq m}Q_j[f]\right\rVert^{p}_{L^p(\mathbb{R})}\right)^{1/p} \\
    &= \left\lVert\sum_{|j| \leq m} Q_j[f]\right\rVert_{L^p(\mathbb{R})}
\end{align*}
Using the density of $L^2(\mathbb{R}) \cap L^p(\mathbb{R}$ in $L^p(\mathbb{R})$ and the completeness of the Haar basis, we find that 
$$\lVert S^{r, \beta}_{m}(f)\rVert_{L^p(\mathbb{R})} \lesssim_{p} \lVert f \rVert_{L^p(\mathbb{R})}$$
The other side can be deduced by a duality argument. Take for example $g = \frac{|f|^p}{f}$, and approximate them by $\varphi_n \in L^2(\mathbb{R}) \cap L^p(\mathbb{R})$ and $\psi_n \in L^2(\mathbb{R} \cap L^{p'}(\mathbb{R})$ respectively to get
\begin{align*}
\lVert f \rVert_{L^p(\mathbb{R})}^{p} &= \int_{\mathbb{R}}f(x)g(x)\,dx \\
&= \lim_{n \to \infty} \int_{\mathbb{R}}\varphi_n(x) \psi_n(x)\,dx \\
&= \lim_{n \to \infty} \sum_{I \in r\mathcal{D}^{\beta}}\langle \varphi_n, h_I \rangle \langle \psi_n, h_I \rangle \\
&= \lim_{n \to \infty} \int_{\mathbb{R}} \sum_{I \in \mathcal{D}^{r, \beta}}(\langle \varphi_n, h_I \rangle h_I(x))(\langle \psi_n, h_I\rangle h_I(x))\,dx \\
&\leq \lim_{n \to \infty} \int_{\mathbb{R}}S^d(\varphi_n)(x)S^d(\psi_n)(x)\,dx \\
&\leq \lim_{n \to \infty} \lVert S^d(\varphi_n) \rVert_{L^p(\mathbb{R})}\lVert S^d(\psi_n)\rVert_{L^{p'}(\mathbb{R})} \\
&\lesssim_{p} \lVert S^{r, \beta}(f)\rVert_{L^p(\mathbb{R})}\lVert f \rVert_{L^p(\mathbb{R})}^{p - 1}
\end{align*}
