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${l}^{q}$-valued extension of the fractional maximal operators for non-doubling measures via potential operators. (English) Zbl 1096.42007

Let $\mu$ be a Radon measure on ${R}^{d}$ and $B\left(x,r\right)$ be a ball with center $x$ and radius $r$. The fractional maximal operator is defined as

${M}_{\alpha }f\left(x\right)=\underset{r>0}{sup}\frac{1}{\mu {\left(B\left(x,r\right)\right)}^{1-\alpha }}{\int }_{B\left(x,r\right)}|f\left(y\right)|\phantom{\rule{0.166667em}{0ex}}d\mu \left(y\right)\phantom{\rule{1.em}{0ex}}0<\alpha <1·$

When $\mu$ satisfies the growth condition $\mu \left(B\left(x,r\right)\right)\le C{r}^{n}$, J. García-Cuerva and A. E. Gatto [Stud. Math. 162, No. 3, 245–261 (2004; Zbl 1045.42006)] defined the following fractional operator

${I}_{\alpha }f\left(x\right)={\int }_{{R}^{d}}\frac{f\left(y\right)}{{|x-y|}^{n-\alpha }}\phantom{\rule{0.166667em}{0ex}}d\mu \left(y\right),$

and obtained ${L}^{p}\left(\mu \right)\to {L}^{q}\left(\mu \right)$ boundedness of ${I}_{\alpha }$ where $1/q=1/p-\alpha /n$.

Without assuming the growth condition on $\mu$, the author considers some potential-like operator ${J}_{\alpha }$ which satisfies the following:

${M}_{\alpha }f\left(x\right)\le C{J}_{\alpha }|f|\left(x\right),\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}{J}_{\alpha }\phantom{\rule{4pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{bounded}\phantom{\rule{4.pt}{0ex}}\text{from}\phantom{\rule{4pt}{0ex}}{L}^{p}\left(\mu \right)\phantom{\rule{4pt}{0ex}}\text{to}\phantom{\rule{4pt}{0ex}}{L}^{q}\left(\mu \right)\phantom{\rule{1.em}{0ex}}1/q=1/p-\alpha ·$

${J}_{\alpha }$ is defined as follows: Let ${r}_{k}\left(x\right)=sup\left\{r\ge 0;\mu \left(B\left(x,r\right)\right)<{2}^{k}\right\}$ for $k\in Z$ with $k>{log}_{2}\mu \left(\left\{x\right\}\right)$.

${J}_{\alpha }f\left(x\right)=\sum _{k=\left[{log}_{2}\mu \left(\left\{x\right\}\right)\right]+1}^{\infty }\frac{1}{{2}^{k\left(1-\alpha \right)}}{\int }_{B\left(x,{r}_{k}\left(x\right)\right)}f\left(y\right)\phantom{\rule{0.166667em}{0ex}}d\mu \left(y\right)·$

The author also considers some vector-valued inequalities of Fefferman-Stein type, uncentered maximal functions, and the boundedness on Morrey spaces.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 26A33 Fractional derivatives and integrals (real functions) 42B35 Function spaces arising in harmonic analysis