Abstract
This paper presents a weak-(p, q) inequality for fractional integral operator on Morrey spaces over metric measure spaces of nonhomogeneous type. Both parameters p and q are greater than or equal to one. The weak-(p, q) inequality is proved by employing an inequality involving maximal operator on the spaces under consideration.
Highlights
Fractional integral operator, which is firstly defined by (Hardy and Littlewood, 1927), is an inverse for a power of the Laplacian operator on Euclidean spaces
It can be found that the works on fractional integral operator have been developed in some directions
Some properties of maximal operator M can be found for example in (Terasawa, 2006)
Summary
Fractional integral operator, which is firstly defined by (Hardy and Littlewood, 1927), is an inverse for a power of the Laplacian operator on Euclidean spaces. This operator is called the Riesz potential. A metric measure space (X, d, μ) is a homogeneous type space if the Borel measure μ satisfies the doubling condition, that is there exists a positive constant C such that for every ball B(a, r) the condition μ(B(x, 2r)) ≤ Cμ(B(x, r)). If the doubling condition does not hold, we have a metric measure space of nonhomogeneous type. The action on the spaces of nonhomogeneous type goes back to the works of (Nazarov, et al, 1998), (Tolsa, 1998), and (Verdera, 2002)
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