Schauder estimates for equations with fractional derivatives

The equation (*) D-t(alpha)(u - h(1)) + D-x(beta)(u - h(2)) = f, 0 < alpha, beta < 1, t, x greater than or equal to 0, where D-t(alpha) and D-x(beta) are fractional derivatives of order alpha and beta is studied. It is shown that if f = f((t) over bar,(x) over bar), h(1) = h(1)((x) over bar), and h(2) = h(2)((t) over bar) are Holder-continuous and f(0, 0) = 0, then there is a solution such that D(t)(alpha)u and D(x)(beta)u are Holder-continuous as well. This is proved by first considering an abstract fractional evolution equation and then applying the results obtained to (*). Finally the solution of (*) with f = 1 is studied.