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Jukka Kohonen
Jukka Kohonen
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Oscillation and HolderHölder continuity

Where can I find a proof of the follwingfollowing fact?

If $$w(u,x_{0},r)=\sup _{B_{r}(x_{0})}u-\inf _{B_{r}(x_{0})}u$$ for some function $u(x)$ satisfies $$ w\left(u,x_{0},{\tfrac {r}{2}}\right)\leq \lambda w\left(u,x_{0},r\right)$$ for a fixed $0 < \lambda < 1$ and all sufficiently small values of $r$, then $u$ is HolderHölder continuous.

Oscillation and Holder continuity

Where can I find a proof of the follwing fact?

If $$w(u,x_{0},r)=\sup _{B_{r}(x_{0})}u-\inf _{B_{r}(x_{0})}u$$ for some function $u(x)$ satisfies $$ w\left(u,x_{0},{\tfrac {r}{2}}\right)\leq \lambda w\left(u,x_{0},r\right)$$ for a fixed $0 < \lambda < 1$ and all sufficiently small values of $r$, then $u$ is Holder continuous.

Oscillation and Hölder continuity

Where can I find a proof of the following fact?

If $$w(u,x_{0},r)=\sup _{B_{r}(x_{0})}u-\inf _{B_{r}(x_{0})}u$$ for some function $u(x)$ satisfies $$ w\left(u,x_{0},{\tfrac {r}{2}}\right)\leq \lambda w\left(u,x_{0},r\right)$$ for a fixed $0 < \lambda < 1$ and all sufficiently small values of $r$, then $u$ is Hölder continuous.

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user60665
user60665

Oscillation and Holder continuity

Where can I find a proof of the follwing fact?

If $$w(u,x_{0},r)=\sup _{B_{r}(x_{0})}u-\inf _{B_{r}(x_{0})}u$$ for some function $u(x)$ satisfies $$ w\left(u,x_{0},{\tfrac {r}{2}}\right)\leq \lambda w\left(u,x_{0},r\right)$$ for a fixed $0 < \lambda < 1$ and all sufficiently small values of $r$, then $u$ is Holder continuous.