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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 \[ M_{\alpha}f(x) = \sup_{r>0} \frac{1}{\mu (B(x,r))^{1- \alpha}} \int_{B(x,r)} | f(y) | \,d \mu (y) \quad 0 < \alpha <1. \] When \(\mu\) satisfies the growth condition \(\mu (B(x,r)) \leq C r^n\), J. GarcĂ­a-Cuerva and A. E. Gatto [Stud. Math. 162, No. 3, 245–261 (2004; Zbl 1045.42006)] defined the following fractional operator \[ I_{\alpha}f(x) = \int_{R^d} \frac{f(y)}{ | x-y |^{n- \alpha}} \,d \mu (y), \] 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: \[ M_{\alpha}f (x) \leq C J_{\alpha} | f | (x) , \quad \text{and}\quad J_{\alpha} \;\text{is bounded from} \;L^p (\mu ) \;\text{to} \;L^{q} (\mu ) \quad 1/q = 1/p - \alpha . \] \(J_{\alpha}\) 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.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
26A33 Fractional derivatives and integrals
42B35 Function spaces arising in harmonic analysis

Citations:

Zbl 1045.42006
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