ACTA issues

## An equivalent norm on BMO spaces

Stevo SteviÄ‡

Acta Sci. Math. (Szeged) 66:3-4(2000), 553-563
2751/2009

 Abstract. Let $p>0$. A Borel function $f$, locally integrable in the unit ball $B$, is said to be a $BMO_p(B)$ function if $$||f||_{BMO_p}=\sup_{B(a,r)\subset B}\big(\frac{1}{V(B(a,r))}\int_{B(a,r)}|f(x)-f_{B(a,r)}|^pdV(x)\big)^{1/p}<+\infty,$$ where the supremum is taken over all balls $B(a,r)$ in $B$, and $f_{B(a,r)}$ is the mean value of $f$ over $B(a,r)$. Let ${\cal H}(B)$ denote the set of harmonic functions in open unit ball $B$, $f_{a,r}(x)$ denotes $f(a+rx)$ for arbitrary function $f$. The main result of this paper is to prove the following theorem: Let $u\in{\cal H}(B)$, $p>1$. Then a) \eqalign{||u||_{BMO_p}^p=\sup_{{a\in B}\atop{0< r< 1-|a|}}\frac{p(p-1)}{2n(n-2)} \int_B\big(&|u_{a,r}(x)-u_{a,r}(0)|^{p-2} |\nabla u_{a,r}(x)|^2\times\cr &\times(2|x|^{2-n}+(n-2)|x|^2-n)\big)dV_N(x)} for $n\geq3$, and b) \eqalign{||u||_{BMO_p}^p=\sup_{{a\in B}\atop{0< r< 1-|a|}}p(p-1)\int_B\big(&|u_{a,r}(x)-u_{a,r}(0)|^{p-2} |\nabla u_{a,r}(x)|^2\times\cr &\times\big(\ln\frac {1}{|x|}-1+|x|\big)\big)dV_N(x)} for $n=2$. AMS Subject Classification (1991): 31B05, 31C05 Received March 8, 1999. (Registered under 2751/2009.)