\u660e\u663e$L^\\infty\\subset BMO$, \u4e3a\u4e86\u8bf4\u660e\u53cd\u4e4b\u4e0d\u4e00\u5b9a\u6210\u7acb, \u6211\u4eec\u9a8c\u8bc1$u(x)=\\log(x)$, $x\\in(0,+\\infty)$\u662f\u5c5e\u4e8eBMO\u7684.<\/p>\n
\u6309\u7167BMO\u7a7a\u95f4\u7684\u5b9a\u4e49, \u9700\u8981\u8bc1\u660e
\n$$
\n[u]_{x,r}\\eqdef\\frac{1}{2r}\\int_{x-r}^{x+r}|u(y)-u_{x,r}|dy< +\\infty,\\quad\\forall 0 < r < x. $$ \u8fd9\u91cc$u_{x,r}\\eqdef\\frac{1}{2r}\\int_{x-r}^{x+r}\\log ydy$.
\n<\/p>\n
\u9996\u5148, \u6ce8\u610f\u5230, \u7531$\\log$\u7684\u6027\u8d28, \u5bf9$\\lambda>0$,
\n\\begin{align*}
\nu_{\\lambda x,\\lambda r}&=\\frac{1}{2\\lambda r}\\int_{\\lambda x-\\lambda r}^{\\lambda x+\\lambda r}\\log y dy\\\\
\n&=\\frac{1}{2r}\\int_{x-r}^{x+r}(\\log y+\\log\\lambda )dy\\\\
\n&=u_{x,r}+\\log\\lambda.
\n\\end{align*}
\n\u56e0\u6b64
\n\\begin{align*}
\n[u]_{\\lambda x,\\lambda r}&=\\frac{1}{2\\lambda r}\\int_{\\lambda x-\\lambda r}^{\\lambda x+\\lambda r}|u(y)-u_{\\lambda x,\\lambda r}|dy\\\\
\n&=\\frac{1}{2r}\\int_{x-r}^{x+r}|u(\\lambda y)-u_{x,r}-\\log\\lambda |dy\\\\
\n&=[u]_{x,r}.
\n\\end{align*}
\n\u53ef\u89c1, \u6211\u4eec\u53ea\u9700\u8ba1\u7b97$\\sup_{x\\in(1,+\\infty)}[u]_{x,1}<+\\infty$\u5373\u53ef.<\/p>\n
\u6ce8\u610f\u5230
\n\\begin{align*}
\nu_{y,1}&=\\frac{1}{2}\\int_{y-1}^{y+1}\\log ydy
\n=\\frac{1}{2}(x\\log x-x)|_{y-1}^{y+1}\\\\
\n&=1+\\frac{1}{2}\\log(y^2-1)+\\frac{1}{2}y\\log\\frac{y+1}{y-1},
\n\\end{align*}
\n\u6bd4\u8f83\u590d\u6742. \u6211\u4eec\u9700\u8981\u5982\u4e0b\u5f15\u7406
\n