Sobolev estimates for fractional parabolic equations with space-time non-local operators

We obtain Lp estimates for fractional parabolic equations with space-time non-local operators partial derivative(alpha)(t)u - Lu + lambda u = f in (0, T) x R-d, where partial derivative(alpha)(t)u is the Caputo fractional derivative of order alpha is an element of (0, 1], T is an element of (0, infinity), and Lu(t, x):= integral(Rd) (u(t, x + y) - u(t, x) - y middot del(x)u(t, x)chi((sigma))(y)) K(t, x, y) dy is an integro-differential operator in the spatial variables. Here we do not impose any regularity assumption on the kernel K with respect to t and y. We also derive a weighted mixed-norm estimate for the equations with operators that are local in time, i.e., alpha = 1, which extend the previous results in Mikulevicius and Pragarauskas (J Differ Equ 256(4):1581-1626, 2014) and Zhang (Annales l'IHPAnalyse Nonlin & eacute;aire 30:573-614, 2013) by using a quite different method.