Smoothness of the remainder in the Littlewood-Paley theory

Question

Assume $f\in L^1(\mathbb{R}^d)$ and $\eta\in C_c^\infty(\mathbb{R}^d)$. What can I know about the smoothness of the non-homogeneous remainders $\mathbf{R}(f,\eta):=\sum_{|j-k|\le 2}\Delta_j f\Delta_k \eta$, $\mathbf{T}_{lh}(f,\eta):=\sum_{j\ge -1}S_{j-2} f\Delta_j \eta$ and $\mathbf{T}_{hl}(f,\eta):=\sum_{j\ge -1}\Delta_j f S_{j-2}\eta$? Here we do the Littlewood-Paley decomposition as follows, $$ u=\Delta_{-1}u + \sum_{j\ge 0}\Delta_j u, \qquad\widehat{\Delta_{-1}u}=\chi\hat{u}, \quad\widehat{\Delta_{j}u}=\varphi_j\hat{u}. $$ From the definition of Besov spaces $B_{p,r}^s$, we can derive the estimation $\|\mathbf{R}(f,\eta)\|_{B_{p,r}^s}\le_{\eta}\|f\|_{B_{p_0,r_0}^{s_1}}$, where $1\le p\le p_0$, $1\le r\le r_0$ and $s,s_1\le 0$ satisfying that $s_1\le s<s_1+2$. I think that this result may be helpful.