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 $\|f\|\_{C^{1,\alpha}(B\_1(0))} \leq c\|f\|\_{L^2(B\_2(0))}$.



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


