Wang Silei.A NOTE ON MORREY-NIKOLSKII INEQUALITY[J].Chinese Annals of Mathematics B,1984,5(1):73~76
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Authors：
Wang Silei;
Abstract：
Let $\[{Q_0}\]$ be a Cube in $\[{R^n}\]$ and $\[u(x) \in {L^p}({Q_0})\]$. Suppose that
$$\[\int_Q {{{\left| {u(x + t) - u(x)} \right|}^p}dx \le {K^p}{{\left| t \right|}^{\alpha p}}{{\left| Q \right|}^{1 - \beta /n}}} \]$$
for all parallel subcubes Q in $\[{Q_0}\]$ and for all t such that the integral makes sense with $\[K \ge 0,0 < \alpha \le 1,0 \le \beta \le n\]$ and $\[p \ge 1\]$.
If $\[\alpha p = \beta \]$ then $\[u(x)\]$ is of bounded mean oscillation on $\[{Q_0}\]$ (abbreviated to $\[BMO({Q_0})\]$， i.e.
$$\[\mathop {\sup }\limits_{Q \subset {Q_0}} \frac{1}{{\left| Q \right|}}\int_Q {\left| {u(x) - {u_Q}} \right|} dx = {\left\| u \right\|_*} < \infty \]$$
where $\[{{u_Q}}\]$ is the mean value of $\[{u(x)}\]$ over Q.
