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Boundedness of Lusin-area and functions on localized BMO spaces over doubling metric measure spaces. (English) Zbl 1220.42014

Let $\left(X,d\right)$ be a metric space endowed with a regular Borel measure $\mu$ such that $0<\mu \left(B\left(x,r\right)\right)<\infty$ $\left(x\in X$, $r>0\right)$. $\left(X,d,\mu \right)$ is called a doubling metric measure space if $\mu \left(B\left(x,2r\right)\right)\le {C}_{1}\mu \left(B\left(x,r\right)\right)$. A positive function $\rho$ on $X$ is called admissible if, for some ${C}_{0}$, ${k}_{0}>0$, $\rho {\left(x\right)}^{-1}\le {C}_{0}\rho {\left(y\right)}^{-1}{\left(1+\rho {\left(y\right)}^{-1}d\left(x,y\right)\right)}^{{k}_{0}}$ $\left(x,y\in X\right)$. For an admissible function $\rho$ and $q\in \left[1,\infty \right)$, a function $f\in {L}_{\text{loc}}^{q}\left(X\right)$ is said to be in ${\text{BMO}}_{\rho }^{q}\left(X\right)$ if

$\begin{array}{cc}\hfill {\parallel f\parallel }_{{\text{BMO}}_{\rho }^{q}\left(X\right)}=& \underset{B=B\left(x,r\right);\phantom{\rule{0.166667em}{0ex}}x\in X,\phantom{\rule{0.166667em}{0ex}}r<\rho \left(x\right)}{sup}{\left(\mu {\left(B\right)}^{-1}{\int }_{B}{|f\left(y\right)-{f}_{B}|}^{q}\phantom{\rule{0.166667em}{0ex}}d\mu \left(y\right)\right)}^{1/q}\hfill \\ \hfill +& \underset{B=B\left(x,r\right);\phantom{\rule{0.166667em}{0ex}}x\in X,\phantom{\rule{0.166667em}{0ex}}r\ge \rho \left(x\right)}{sup}{\left(\mu {\left(B\right)}^{-1}{\int }_{B}{|f\left(y\right)|}^{q}\phantom{\rule{0.166667em}{0ex}}d\mu \left(y\right)\right)}^{1/q}<\infty ·\hfill \end{array}$

Similarly, a real valued function $f\in {L}_{\text{loc}}^{q}\left(X\right)$ is said to be in ${\text{BLO}}_{\rho }^{q}\left(X\right)$ if, in the definition of ${\text{BMO}}_{\rho }^{q}\left(X\right)$, ${f}_{B}$ is replaced by $\text{ess}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4.pt}{0ex}}{\text{inf}}_{u\in B}f\left(u\right)$. In the case $\left(X,d,\mu \right)=\left({ℝ}^{n},|·|,dx\right)$ and $\rho \equiv 1$, ${\text{BMO}}_{\rho }^{q}\left(X\right)$ coincides with the Goldberg’s local BMO space bmo. If $\rho$ is an admissible function, ${\text{BMO}}_{\rho }^{q}\left(X\right)\cong {\text{BMO}}_{\rho }^{1}\left(X\right)=:{\text{BMO}}_{\rho }\left(X\right)$.

Now let $\rho$ be an admissible function and ${\left\{{Q}_{t}\right\}}_{t\ge 0}$ be a family of operators bounded on ${L}^{2}\left(X\right)$ with integral kernels ${\left\{{Q}_{t}\left(x,y\right)\right\}}_{t\ge 0}$ satisfying that there exist $C$, ${\delta }_{1}>0$, ${\delta }_{2}\in \left(0,1\right)$, $\gamma >0$ such that $|{Q}_{t}\left(x,y\right)|\le C\left(\mu \left(B\left(x,t\right)\right)+\mu {\left(B\left(x,d\left(x,y\right)\right)\right)}^{-1}{\left(1+d\left(x,y\right)/t\right)}^{-\gamma }{\left(1+t/\rho \left(x\right)\right)}^{-{\delta }_{1}}$, and $|{\int }_{X}{Q}_{t}{\left(x,z\right)\phantom{\rule{0.166667em}{0ex}}d\mu \left(z\right)|\le C\left(1+\rho \left(x\right)/t\right)}^{-{\delta }_{2}}$. Using this family of operators, the authors define the Littlewood-Paley $g$-function $g\left(f\right)$, Lusin’s area function $S\left(f\right)$ and the ${g}_{\lambda }^{*}$ function ${g}_{\lambda }^{*}\left(f\right)$ by

$\begin{array}{cc}\hfill g\left(f\right)\left(x\right)& ={\left({\int }_{0}^{\infty }{|{Q}_{t}\left(f\right)\left(x\right)|}^{2}\phantom{\rule{0.166667em}{0ex}}dt/t\right)}^{1/2},\hfill \\ \hfill S\left(f\right)\left(x\right)& ={\left({\int }_{0}^{\infty }{\int }_{d\left(x,y\right)

Their main result is the following: Let $X$ be a doubling metric measure space satisfying one more condition, “the $\delta$-annular decay property”. Let $\rho$ be an admissible function. Assume that the Littlewood-Paley $g$-function is bounded on ${L}^{2}\left(X\right)$. Then ${\parallel S\left(f\right)}^{2}{\parallel }_{{\text{BLO}}_{\rho }\left(X\right)}\le C{\parallel f\parallel }_{{\text{BMO}}_{\rho }\left(X\right)}^{2}$.

If $3n<\lambda <\infty$, the same result holds for the ${g}_{\lambda }^{*}$ function without assuming “the $\delta$-annular decay property”.

Relating to their boundedness results, the authors give a nonnegative $f\in \text{bmo}\left(ℝ\right)$ which is not in $\text{blo}\left(ℝ\right)$.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 42B30 ${H}^{p}$-spaces (Fourier analysis) 51F99 Metric geometry
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