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BMOEstimates for Maximal Operators via Approximations of the Identity with NonDoubling Measures


Published:20100729
Printed: Dec 2010
Dachun Yang,
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex systems, Ministry of Education, People's Republic of China
Dongyong Yang,
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex systems, Ministry of Education, People's Republic of China
Abstract
Let $\mu$ be a nonnegative Radon measure
on $\mathbb{R}^d$ that satisfies the growth condition that there exist
constants $C_0>0$ and $n\in(0,d]$ such that for all $x\in\mathbb{R}^d$ and
$r>0$, ${\mu(B(x,\,r))\le C_0r^n}$, where $B(x,r)$ is the open ball
centered at $x$ and having radius $r$. In this paper, the authors prove
that if $f$ belongs to the $\textrm {BMO}$type space $\textrm{RBMO}(\mu)$ of Tolsa, then
the homogeneous maximal function $\dot{\mathcal{M}}_S(f)$ (when $\mathbb{R}^d$ is not an
initial cube) and the inhomogeneous maximal function
$\mathcal{M}_S(f)$ (when $\mathbb{R}^d$ is an initial cube)
associated with a given approximation of the identity $S$ of Tolsa are
either infinite everywhere or finite almost everywhere,
and in the latter case, $\dot{\mathcal{M}}_S$ and $\mathcal{M}_S$ are bounded from
$\textrm{RBMO}(\mu)$ to the $\textrm {BLO}$type
space $\textrm{RBLO}(\mu)$. The authors also prove that the inhomogeneous
maximal operator $\mathcal{M}_S$ is bounded from the local
$\textrm {BMO}$type space $\textrm{rbmo}(\mu)$
to the local $\textrm {BLO}$type space $\textrm{rblo}(\mu)$.
Keywords: 
Nondoubling measure, maximal operator, approximation of the identity, RBMO(mu), RBLO(mu), rbmo(mu), rblo(mu)
Nondoubling measure, maximal operator, approximation of the identity, RBMO(mu), RBLO(mu), rbmo(mu), rblo(mu)

MSC Classifications: 
42B25, 42B30, 47A30, 43A99 show english descriptions
Maximal functions, LittlewoodPaley theory $H^p$spaces Norms (inequalities, more than one norm, etc.) None of the above, but in this section
42B25  Maximal functions, LittlewoodPaley theory 42B30  $H^p$spaces 47A30  Norms (inequalities, more than one norm, etc.) 43A99  None of the above, but in this section
