Speaker
Description
We consider maximal kernel-operators on measure spaces $(X,\mu)$ equipped with a ball-basis. We prove that under certain asymptotic condition on the kernels those operators maps boundedly $\text{BMO(X)}$ into $\text{BLO(X)}$, 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
$$\mathcal{M}_{\phi} f(x)=\sup_{r>0}\frac{1}{r^d}\int_{\mathbb{R}^d}|f(t)|\phi\left(\frac{x-t}{r}\right)dt,$$
and its non-tangential version, where $\phi(x)\ge 0$ is a bounded, integrable spherical function on $\mathbb{R}^d$, decreasing with respect to $|x|$. We prove that $\mathcal{M}_\phi$ is bounded from $\text{BMO}(\mathbb{R}^d)$ to $\text{BLO}(\mathbb{R}^d)$ if and only if
$$\int_{\mathbb{R}^d}\phi(x)\log (2+|x|)dx<\infty. $$
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 $f$.
