Mean Oscillation Over Cubes and Hölder Continuity

\mathbf{Problem:} Suppose f\in L^1_{Loc}(\mathbb{R}^n) is hölder \alpha-continuous, where 0<\alpha\leq 1. It is equivalent to say for any ball B centered at x\in \mathbb{R}^n we have

\displaystyle \frac{1}{|B|}\int_B\left|f(y)-\frac{1}{|B|}\int_B f(z)dz\right|dy\leq C|B|^{\alpha/n}

where C is independent from x and the radius of the ball.

\mathbf{Proof:} “necessity” If f is hölder \alpha-continuous, then

\displaystyle \frac{1}{|B|}\int_B\left|f(y)-\frac{1}{|B|}\int_B f(z)dz\right|dy\leq \frac{1}{|B|^2}\int_B\int_B|f(y)-f(z)|dydz

\displaystyle \leq \frac{C}{|B|^2}\int_B\int_B |y-z|^\alpha dydz\leq \frac{C}{|B|^2}\int_B\int_B(2r)^\alpha dydz=Cr^\alpha=C|B|^{\alpha/n}

here r is the radius of B.

“sufficiency” Let B_k=B(x_0,2^{-k}r) is the ball centered at x_0 with radius 2^{-k}r. Denote \displaystyle f_B=\frac{1}{|B|}\int_Bf(y)dy then \displaystyle\frac{1}{|B|}\int_B|f(y)-f_B|dy\leq C|B|^{\alpha/n}.

Since \displaystyle |f_{B_k}-f_{B_{k+1}}|\leq |f(y)-f_{B_k}|+|f(y)-f_{B_{k+1}}|, \forall\, y\in B_{k+1}

Integrating this over B_{k+1}, we get

\displaystyle |f_{B_k}-f_{B_{k+1}}|\leq \frac{1}{|B_{k+1}|}\int_{B_{k+1}}|f(y)-f_{B_{k+1}}|dy+\frac{|B_k|}{|B_{k+1}|}\frac{1}{|B_k|}\int_{B_k}|f(y)-f_{B_k}|dy\leq C|B_{k+1}|^{\alpha/n}+C2^n|B_k|^{\alpha/n}\leq C(n)2^{-k\alpha}r^\alpha

So \displaystyle |f_{B_{k}}-f_{B_{k+1}}|\leq \sum\limits_{i=k}^{k+l-1}|f_{B_i}-f_{B_{i+1}}|\leq \sum\limits_{i=k}^{k+l-1}C(n)2^{-i\alpha}r^\alpha\leq C(n)r^\alpha 2^{-(k-1)\alpha}.

Since f\in L^1_{Loc}(\mathbb{R}^n), from lebesgue differentiation theorem \lim\limits_{n\to\infty}f_{B_{m}}=f(x) a.e. Let l\to \infty, we know |f_{B_k}-f_{x_0}|\leq C(n)2^\alpha r^\alpha a.e. k\geq 0.

So when k=0, |f_{B_0}-f(x_0)|\leq C(n)2^\alpha r^\alpha.

Set r=|x_0-y_0|, B'_0=B(y_0, r), then

\displaystyle |f(x_0)-f(y_0)|\leq |f(x_0)-f_{B_0}|+|f(y_0)-f_{B'_0}|+|f_{B_0}-f_{B'_0}|\leq C(n)2^{\alpha+1}r^\alpha+|f_{B_0}-f_{B'_0}| holds almost everywhere.

Since we have |f_{B_0}-f_{B'_0}|\leq |f(z)-f_{B_0}|+|f(z)-f_{B'_0}|, z\in B_0\cap B'_0

Integrating this on B_0\cap B'_0, we get

\displaystyle |f_{B_0}-f_{B'_0}|\leq \frac{|B_0|}{|B_0\cap B'_0|}\frac{1}{|B_0|}\int |f(z)-f_{B_0}|dz+\frac{|B'_0|}{|B_0\cap B'_0|}\frac{1}{|B'_0|}\int|f(z)-f_{B'_0}|dz\leq C(n)|B_0|^{\alpha/n}+C(n)|B'_0|^{\alpha/n}=C(n)r^\alpha

So |f(x_0)-f(y_0)|\leq C(n,\alpha)r^\alpha=C(n,\alpha)|x_0-y_0|^\alpha a.e. So f is a hölder \alpha-continuous.

\text{Q.E.D}\hfill \square\mathbf{Remark:} Fanghua Lin, Qing Han. Elliptic partial differential equations.

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