In [Estimates of fractional heat kernel](https://mathoverflow.net/questions/382309/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 $\nabla\_x p\_t^{(n)}(x) \in L^p((0,T); L^q(\mathbb R^n))$?



Integrability of fractional heat kernel
Of course we can!




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And if we actually do, then we find that $\phi(t, x) = |\nabla\_x p\_t^{(n)}(x)|$ satisfies $$\phi(t, x) = t^{-(n+1)/(2s)} \phi(1, t^{-1/(2s)} x)$$ and, with $\tilde x = (x, 0, 0) \in \mathbb R^{n+2}$), $$\phi(1,x) = |x| p\_1^{(n+2)}(\tilde x) \approx C |x| \min\{1, |x|^{-(n+2+2s)}\} = C \min\{|x|, |x|^{-(n+1+2s)}\} $$ (here we use the well-known fact that $p\_1^{(n)}(x) \approx C \min\{1, |x|^{-n-2s}\}$). It follows that $$\|\phi(1,\cdot)\|\_q < \infty \qquad \text{if and only if } q > \tfrac{n}{n+1+2s} $$ and $$\|\phi(t,\cdot)\|\_q = t^{-(n+1)/2s + n/(2qs)} ,$$ and the above is in $L^p([0,T])$ if and only if $$ p (\tfrac{n+1}{2s} - \tfrac{n}{2qs}) < 1 . $$
So the answer seems to be rather ugly:
$$ q > \tfrac{n}{n+1+2s} \qquad \text{and} \qquad p (\tfrac{n+1}{2s} - \tfrac{n}{2qs}) < 1 , $$
unless I made some mistake.


