Maximal operators on spaces BMO and BLO

We consider maximal kernel-operators on abstract measure spaces $(X,\mu)$ equipped with a ball-basis. We prove that under certain asymptotic condition on the kernels those operators maps boundedly BMO(X) into BLO(X), generalizing the well-known results of Bennett-DeVore-Sharpley and Bennett for the Hardy-Littlewood maximal function. As a particular case of such an operator one can consider the maximal function \begin{equation} M_\phi f(x)=\sup_{r>0}\frac{1}{r^d}\int_{R^d}|f(t)|\phi\left(\frac{x-t}{r}\right)dt, \end{equation} and its non-tangential version. Here $\phi(x)\ge 0$ is a bounded spherical function on $R^d$, decreasing with respect to $|x|$ and satisfying the bound \begin{equation*} \int_{R^d}\phi (x)\log (2+|x|)dx<\infty. \end{equation*} We prove that if $f\in BMO(R^d)$ and $M_\phi(f)$ is not identically infinite, then $M_\phi(f)\in BLO(R^d)$. Our main result is an inequality, providing an estimation of certain local oscillation of the maximal function $M(f)$ by a local sharp function of $f$.