$C$ only depends on $d,\delta,K,\alpha,\beta,p, q,T$ and $\kappa_0$.

(ii) If in addition, we also assume that $p>\frac{d(3\beta-1)}{2(\beta-1)}$ and $$\kappa_1:=\sup_{t,\v}\|\Delta_x^{\frac{1}{3}}\sigma_t(\cdot,\v)\|_p+\|b\|_{\mH^{2/3,0}_p(T)}<\infty,$$ then for any $f\in \mH^{2/3,0}_p(T)$, the unique solution $u$ also satisfies $$\begin{aligned}
    \label{Pr1}
    \|\nabla_x\nabla_\v u^\lambda\|_{\mL^p(T)}+\|\Delta^{\frac{1}{3}}_x\nabla^2_\v u^\lambda\|_{\mL^p(T)}\leq C\|f\|_{\mH^{2/3,0}_p(T)},\end{aligned}$$ and for all $t\in[0,T]$, $$\begin{aligned}
    \label{Pr2}
    \|\Delta_x^{\frac{1}{3}}  u^\lambda_t\|_{\alpha,\beta;p}\leq C((T-t)\wedge\lambda^{-1})^{1-\frac{1}{p}-\frac{(3\alpha)\vee\beta}{2}}\|f\|_{\mH^{2/3,0}_p(t,T)},\end{aligned}$$ where the constant $C$ only depends on $d,\delta,K,\alpha,\beta,p, T$ and $\kappa_1$.

In order to emphasize the dependence of the unique solution $u$ on $a,b$ and $T,\lambda, f$, we sometimes denote $u=\cR^{a,b}_{\lambda,T}(f)$.

Case $b=0$
----------

In this subsection we first consider the case of $b=0$ by using the freezing coefficient argument, and show the following basic existence and uniqueness result for equation . \[Th33\] Let $p>1$ and $\alpha\in[0,\frac{2}{3})$, $\beta\in[0,2)$. Suppose [**(H$^{\delta,p}_K$)**]{} holds.

(i) For any $f\in \mL^p(T)$, there exists a unique solution $u=u^\lambda$ to in the sense of Definition \[Def31\] so that $$\begin{aligned}
    \label{Pri}
    \|u^\lambda\|_{\mH^{2/3,2}_p(T)}\leq C_1\|f\|_{\mL^p(T)}.\end{aligned}$$

(ii) If $p>\frac{2}{(2-3\alpha)\wedge(2-\beta)}$, then for all $t\in[0,T]$, $$\begin{aligned}
    \label{Pri2}
    \|u^\lambda_t\|_{\alpha,\beta;p}\leq C_2((T-t)\wedge\lambda^{-1})^{1-\frac{1}{p}-\frac{(3\alpha)\vee\beta}{2}}\|f\|_{\mL^p(t,T)}.\end{aligned}$$

Here $C_1=C_1(d,\delta,K,p,T)$ and $C_2=C_2(d,\delta,K,\alpha,\beta,p,T)$ are increasing with respect to $T$.

We show the apriori estimates and by the freezing coefficient argument. The existence of a solution follows by the standard continuity argument. We divide the proof into five steps.

[**(a)**]{} First of all, we assume that $u\in C([0,T]; H^{\infty,\infty}_p)$ 