Maximal operators on spaces BMO and BLO
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915660458622976 |
|---|---|
| author | Karagulyan, Grigori A. |
| author_facet | Karagulyan, Grigori A. |
| contents | We consider maximal kernel-operators on abstract measure spaces $(X,μ)$ 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_ϕf(x)=\sup_{r>0}\frac{1}{r^d}\int_{R^d}|f(t)|ϕ\left(\frac{x-t}{r}\right)dt, \end{equation} and its non-tangential version. Here $ϕ(x)\ge 0$ is a bounded spherical function on $R^d$, decreasing with respect to $|x|$ and satisfying the bound \begin{equation*}
\int_{R^d}ϕ(x)\log (2+|x|)dx<\infty. \end{equation*} We prove that if $f\in BMO(R^d)$ and $M_ϕ(f)$ is not identically infinite, then $M_ϕ(f)\in BLO(R^d)$. 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 $f$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05882 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maximal operators on spaces BMO and BLO Karagulyan, Grigori A. Classical Analysis and ODEs 42B25, 42B35, 43A85 We consider maximal kernel-operators on abstract measure spaces $(X,μ)$ 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_ϕf(x)=\sup_{r>0}\frac{1}{r^d}\int_{R^d}|f(t)|ϕ\left(\frac{x-t}{r}\right)dt, \end{equation} and its non-tangential version. Here $ϕ(x)\ge 0$ is a bounded spherical function on $R^d$, decreasing with respect to $|x|$ and satisfying the bound \begin{equation*} \int_{R^d}ϕ(x)\log (2+|x|)dx<\infty. \end{equation*} We prove that if $f\in BMO(R^d)$ and $M_ϕ(f)$ is not identically infinite, then $M_ϕ(f)\in BLO(R^d)$. 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 $f$. |
| title | Maximal operators on spaces BMO and BLO |
| topic | Classical Analysis and ODEs 42B25, 42B35, 43A85 |
| url | https://arxiv.org/abs/2502.05882 |