  Based on some elementary estimates for the space-time derivatives of the heat
kernel, we use a bootstrapping approach to establish the optimal decay rates
for the $L^q(\mathbb{R}^d)$ ($1\leq q\leq\infty$, $d\in\mathbb{N}$) norm of the
space-time derivatives of solutions to the (modified) Patlak-Keller-Segel
equations with initial data in $L^1(\mathbb{R}^d)$, which implies the joint
space-time analyticity of solutions. When the $L^1(\mathbb{R}^d)$ norm of the
initial datum is small, the upper bound for the decay estimates is global in
time, which yields a lower bound on the growth rate of the radius of space-time
analyticity in time. As a byproduct, the space analyticity is obtained for any
initial data in $L^1(\mathbb{R}^d)$. The decay estimates and space-time
analyticity are also established for solutions bounded in both space and time
variables. The results can be extended to a more general class of equations,
including the Navier-Stokes equations.
