Prove
Lemma 11 ([Local Schauder estimates]). Under these assumptions the following…Definition 10 ([Weighted local spaces]). Let M be either ^m or the fixed sphere…Proposition
We recall an interpolation proof of the replacement of the supremum term. On three nested interior cylinders, with a gap of parabolic size a , the estimate already proved bounds \|D_zu\|_∞ and \|u_t\|_∞ on the middle cylinder by Ca^ -k (\|u\|_ ∞;outer +|f|_α) . Average u on a preceding cylinder of radius q<a/4 ending at the point being estimated. Its volume is a fixed multiple of q^ d+4 , so |u(z,t)|≤Cq^ -(d+4)/2 \|u\|_ L^2(outer) +C(q+q^4)a^ -k (\|u\|_ ∞;outer +|f|_α). Choose q to be a sufficiently small fixed multiple of a suitable power of a . The last supremum coefficient can be made any prescribed ϑ>0 , at the cost of another fixed inverse power of a on the other terms. Iteration over geometrically decreasing gaps, with ϑ smaller than the corresponding geometric growth factor, bounds the inner supremum by C(\|u\|_ L^2 +|f|_α) . Lemma 11 ([Local Schauder estimates]) on one further nested cylinder proves; finite covering handles U_1 and the time interval. All averaging cylinders use preceding times. First carry this out up to any closed upper face b'<b on which the local norms are finite, then let b' b . The constants use only the fixed lower time buffer. Pairs of points approaching b are included by this limiting argument.