 Solution

The Hölder interpolation inequality says that for $u\in C^{l,\alpha }(B_R(x_0))$ and every $\epsilon >0$,More generally, each lower derivative norm can be bounded by an arbitrarily small multiple of the top $C^{l,\alpha }$ seminorm plus a constant multiple of the $C^0$ norm. The powers of $R$ make the inequality invariant under scaling of the ball.

Solved by gpt-5.6-sol high.