unique local strong solution $(X_t,\dot X_t)$ up to the explosion time $\zeta$.

Let $$\sigma^R_t(z):=\sigma_t(z\chi_R(z)),\ \ b^R_t(z):=b_t(z)\chi_R(z).$$ By the assumptions, one sees that $(\sigma^R, b^R)$ satisfies the conditions of Theorem \[Main\]. Hence, there exists a unique solution to the following SDE: $$\dif Z^R_t=(\dot X^R_t,b^R_t(Z^R_t))\dif t+(0,\sigma^R_t(Z^R_t)\dif W_t),\ \ Z^R_0=z=(x,\v)\in\mR^{2d},$$ where $Z^R_t=(X^R_t,\dot X^R_t)$. Define $$\zeta_R:=\inf\Big\{t\geq 0: |Z^R_t|\geq R\Big\},\ \ Z_t:=Z^R_t,\ t\in[0,\zeta_R].$$ Since $Z^{R'}_t|_{[0,\zeta_R]}=Z^R_t|_{[0,\zeta_R]}$ for $R'>R$, one sees that $R\mapsto \zeta_R$ is increasing and the above $Z_t$ is well-defined. Clearly, $\zeta=\lim_{R\to\infty}\zeta_R$ is the explosion time of $Z_t$, and $Z_t$ uniquely solves before $\zeta$.

The strategy of proving Theorem \[Main\] is still based on Zvonkin’s transformation. As in the non-degenerate case [@Zh4], we need to establish the $L^p$-maximal regularity estimate to the following degenerate parabolic equation (see Theorem \[Th32\] below): $$\p_t u=\sL^{a,b}_t u+f,\ \ u_0=0.$$ Here we shall use the freezing coefficient argument and the $L^p$-estimate established in [@Br-Cu-La-Pr] and [@Bo] for degenerate operators with constant coefficients (see also [@Ch-Zh] for the case of nonlocal operators). Compared with [@Br-Cu-La-Pr] and [@Pr], we not only consider the optimal regularity of $u$ along the nondegenerate $\v$-direction, but also the optimal regularity of $u$ along the degenerate $x$-direction.

On the other hand, from the viewpoint of PDE, the 