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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 $B(x,r)$ be a ball with center $x$ and radius $r$. The fractional maximal operator is defined as $$ M_{\alpha}f(x) = \sup_{r>0} \frac{1}{\mu (B(x,r))^{1- \alpha}} \int_{B(x,r)} \vert f(y) \vert \,d \mu (y) \quad 0 < \alpha <1. $$ When $\mu$ satisfies the growth condition $\mu (B(x,r)) \leq C r^n$, {\it J. GarcĂ­a-Cuerva} and {\it A. E. Gatto} [Stud. Math. 162, No. 3, 245--261 (2004; Zbl 1045.42006)] defined the following fractional operator $$ I_{\alpha}f(x) = \int_{R^d} \frac{f(y)}{ \vert x-y \vert^{n- \alpha}} \,d \mu (y), $$ and obtained $L^p (\mu ) \to L^{q} (\mu )$ 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: $$ M_{\alpha}f (x) \leq C J_{\alpha} \vert f \vert (x) , \quad \text{and}\quad J_{\alpha} \ \text{is bounded from} \ L^p (\mu ) \ \text{to} \ L^{q} (\mu ) \quad 1/q = 1/p - \alpha . $$ $J_{\alpha}$ is defined as follows: Let $ r_k (x) = \sup\{ r \geq 0 ; \mu (B(x,r)) < 2^k\} $ for $k \in Z$ with $k > \log_2 \mu (\{ x\})$. $$ J_{\alpha}f(x) = \sum_{ k = [ \log_2 \mu ( \{ x \} )] +1 }^{\infty} \frac{1}{2^{k(1 - \alpha)}} \int_{B(x, r_{k}(x))} f(y) \,d \mu (y). $$ The author also considers some vector-valued inequalities of Fefferman-Stein type, uncentered maximal functions, and the boundedness on Morrey spaces.

42B25Maximal functions, Littlewood-Paley theory
26A33Fractional derivatives and integrals (real functions)
42B35Function spaces arising in harmonic analysis