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 \(\textrm{BMO }(X)\) into \(\textrm{BLO }(X)\) , generalizing the well-known results of Bennett-DeVore-Sharpley [2] and Bennett [1] for the Hardy-Littlewood maximal function. As a particular case of such an operator one can consider the maximal function \(\begin{aligned} {\mathcal {M}}_\phi f(x)=\sup _{r>0}\frac{1}{r^d}\int _{\mathbb R^d}|f(t)|\phi \left( \frac{x-t}{r}\right) dt, \end{aligned}\) and its non-tangential version. Here \(\phi (x)\ge 0\) is a bounded spherical function on \(\mathbb R^d\) , decreasing with respect to |x| and satisfying the bound \(\begin{aligned} \int _{\mathbb R^d}\phi (x)\log (2+|x|)dx<\infty . \end{aligned}\) We prove that if \(f\in \textrm{BMO }(\mathbb R^d)\) and \({\mathcal {M}}_\phi (f)\) is not identically infinite, then \({\mathcal {M}}_\phi (f)\in \textrm{BLO }(\mathbb R^d)\) . Our main result is an inequality, providing an estimation of certain local oscillation of the maximal function \({\mathcal {M}}(f)\) by a local sharp function of f.
