A note on Morrey-Nikolskii inequality (Q795281)

In this paper, the following theorem is proved: Let \(Q_ 0\) be a cube in \({\mathbb{R}}^ n\) and \(u(x)\in L^ p(Q_ 0).\) Suppose that \(\int_{Q}| u(x+t)-u(x)|^ pdx\leq K^ p| t|^{\alpha p}| Q|^{1-\beta /n}\) for all parallel subcubes Q of \(Q_ 0\) and for all t such that the integral is defined. The constants are assumed to satisfy \(K\leq 0\), \(0<\alpha \leq 1\), 0\(\leq \beta \leq n\) and \(p\geq 1\). If \(\alpha p=\beta\), then u(x) is of bounded mean oscillation on \(Q_ 0\), i.e., \(\sup_{Q\subset Q_ 0}(1/| Q|)\int_{Q}| u(x)-u_ Q| dx=\| u\|_*<\infty\) where Q is any parallel subcube in \(Q_ 0\), and \(u_ Q\) is the mean value of u(x) over Q. This theorem can be considered as a complement of a theorem established by \textit{J. Ross} [Proc. Am. Math. Soc. 78, 97-102 (1980; Zbl 0453.46026)].