\alpha<3$, and, either $\mu>0, D^\pm\in{\mathbb{R}}$, or $\mu=0$ and $D^- +1<0<D^+$.

First, we consider the case when $\gamma>1/2$. We shall prove boundedness of the second linear operator $\tilde f$. If $u$ is such that $\partial_xu\in X^{\gamma-1/2}$ then there exists $\varphi\in X=L^p({\mathbb{R}})$ such that $\partial_x u = A^{-(2\gamma-1)/2}\varphi  = G_{2\gamma-1} * \varphi$ and $$\Vert \partial_x u\Vert_{X^{\gamma-1/2}} = \Vert \varphi\Vert_X= \Vert \varphi\Vert_{L^p}.$$ Hence, for any $x, \theta$, and $z$ we have $$\frac{\partial u}{\partial x}(x+\theta z)-  \frac{\partial u}{\partial x}(x)
= \left(G_{2\gamma-1}(x+\theta z - \cdot) - G_{2\gamma-1}(x - \cdot) \right)* \varphi(\cdot).$$ Recall the following inequality for the convolution operator: $$\Vert G*\varphi\Vert_{L^p}\le \Vert G\Vert_{L^q} \Vert \varphi\Vert_{L^r},$$ where $p,q,r\ge 1$ and $1/p + 1 = 1/q + 1/r$ (see [@Henry1981 Section 1.6]). In the special case when $q=1$ we have $\Vert G*\varphi\Vert_{L^p}\le \Vert G\Vert_{L^1} \Vert \varphi\Vert_{L^p}$. According to [@Stein1970 Chapter 5.4, Proposition 7] we know that the modulus of continuity of the Bessel kernel function $G_{2\gamma-1}$ satisfies the estimate: $$\Vert G_{2\gamma-1}(\cdot + h) - G_{2\gamma-1}(\cdot) \Vert_{L^1}
\le C_1 |h|^{2\gamma-1},$$ for any $h$ where $C_1>0$ is a constant. Therefore, for any $\theta, z\in{\mathbb{R}}$ we have $$\begin{aligned}
&&\int_{\mathbb{R}}\left|\frac{\partial u}{\partial x}(x+\theta z) -  \frac{\partial u}{\partial x}(x)\right|^p{\mathrm{d}}{x}
= \Vert\left( G_{2\gamma-1}(\cdot + \theta z) - G_{2\gamma-1}(\cdot)\right)*\varphi\Vert_{L^p}^p 
\\
&& \le \Vert G_{2\gamma-1}(\cdot + 