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.