In Estimates of fractional heat kernel, it was stated that $$ \partial_{x_j} p_t^{(n)}(x) = -\frac{x_j}{2 \pi} \, p_t^{(n+2)}(\tilde x) $$ where $x = (x_1, \ldots, x_n) \in \mathbb R^n$, $\tilde x = (x_1, \ldots, x_n, 0, 0) \in \mathbb R^{n+2}$, and $p_t^{(n)}(x)$ is the heat kernel for $(-\Delta)^s$ in $n$ dimensions.

Can we use this information to answer the following question?

**Question:** What are the values of $p$ and $q$ such that $\nabla_x p_t^{(n)}(x) \in L^p((0,T); L^q(\mathbb R^n))$?