Abstract. We give here some equivalent definitions of the so-called $\rho $-Carleson measures when $\rho(t)=(\log(4/t))^p(\log\log (e^4/t))^q$, $0\le p,q< \infty $. As applications, we characterize the pointwise multipliers on $LMOA({\msbm S}^n)$ and from this space to $BMOA({\msbm S}^n)$. Boundedness of the Cesàro type integral operators on $LMOA({\msbm S}^n)$ and from $LMOA({\msbm S}^n)$ to $BMOA({\msbm S}^n)$ is considered as well.

AMS Subject Classification (1991): 28A25; 30D45; 30D50; 47B38

Keyword(s): Carleson measures, Hardy spaces, Bergman spaces, Bloch spaces, BMOA, LMOA

Received September 26, 2008, and in final form January 28, 2009. (Registered under 6426/2009.)