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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 μ 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 α f(x)=sup r>0 1 μ(B(x,r)) 1-α B(x,r) |f(y)|dμ(y)0<α<1·

When μ satisfies the growth condition μ(B(x,r))Cr 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 α f(x)= R d f(y) |x-y| n-α dμ(y),

and obtained L p (μ)L q (μ) boundedness of I α where 1/q=1/p-α/n.

Without assuming the growth condition on μ, the author considers some potential-like operator J α which satisfies the following:

M α f(x)CJ α |f|(x),andJ α isboundedfromL p (μ)toL q (μ)1/q=1/p-α·

J α is defined as follows: Let r k (x)=sup{r0;μ(B(x,r))<2 k } for kZ with k>log 2 μ({x}).

J α f(x)= k=[log 2 μ({x})]+1 1 2 k(1-α) B(x,r k (x)) f(y)dμ(y)·

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

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