Reference for higher order Campanato Lemmas, e.g. `Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}'

Question

Whence can I reference the following fact (I have seen it quoted as `standard' in respectable places, so I hope it is so)?:

Let $f : B_2(0) \to \mathbb{R}$, say $f \in L^2(B_2(0))$ . Suppose that there exists $\alpha \in (0,1)$ and $c > 0$ such that the following is true: For every $y \in B_1(0)$, there exists an affine function $l_y : \mathbb{R}^n \to \mathbb{R}$ such that

$\rho^{-n-2}\int_{B_{\rho}(y)}|f(x) - l_y(x)|^2dx \leq c\rho^{2\alpha}\int_{B_2(0)}|f(x)|^2dx$

for all $\rho \in (0,1/4)$.

Then $f \in C^{1,\alpha}(B_1(0))$ with $\|f\|_{C^{1,\alpha}(B_1(0))} \leq c\|f\|_{L^2(B_2(0))}$.

Answer 1

First you can rescale so $\|f\|_{L^2(B_2)} = 1$ and you see that you are basically talking about the boundedness of the first order Campanato norm of your function $f$. Your conclusion then is a classical theorem concerning the comparison of Campanato and Holder spaces.

See e.g. http://link.springer.com/chapter/10.1007%2F978-3-0348-0537-7_15 (Theorem 4.4) and references therein.