On local regularity estimates for fractional powers of parabolic operators with time-dependent measurable coefficients

We consider fractional operators of the form H-s = (delta(t) - div(x) (A (x, t)del(x)))(s), (x, t) is an element of R-n x R, where s is an element of (0, 1) and A = A (x, t) = {A(i, j)(x, t)}(i,j=1)(n) is an accretive, bounded, complex, measurable, n x n-dimensional matrix valued function. We study the fractional operators H-s and their relation to the initial value problem (lambda(1-2s)u ')'(lambda) = lambda(1-2s)Hu(lambda), lambda is an element of (0, infinity), u(0) = u, in R+ x R-n x R. Exploring the relation, and making the additional assumption that A = A (x, t) = {A(i, j) (x, t)}(i, j=1)(n) is real, we derive some local properties of solutions to the non-local Dirichlet problem H-s u = (delta(t) - div(x) (A(x, t)del(x)))(s) u = 0 for (x, t) is an element of Omega x J, u = f for (x, t) is an element of Rn+1 \ (Omega x J). Our contribution is that we allow for non-symmetric and time-dependent coefficients.