Maximal and fractional integral operators on generalized Morrey spaces over metric measure spaces

Abstract

We establish the boundedness and weak boundedness of the maximal operator and generalized fractional integral operators on generalized Morrey spaces over metric measure spaces (X,d,μ) without the assumption of the growth condition on μ. The results are generalization and improvement of some known results. We also give the vector‐valued boundedness. Moreover we prove the independence of the choice of the parameter in the definition of generalized Morrey spaces by using the geometrically doubling condition in the sense of Hytönen.

A WEAK-(p, q) INEQUALITY FOR FRACTIONAL INTEGRAL OPERATOR ON MORREY SPACES OVER METRIC MEASURE SPACES

“…The fractional integral operator (equation 3) has been proved to satisfy the weak-(1, q) an inequality in Lebesgue spaces over metric measure spaces of nonhomogeneous type (García-Cuerva and Gatto, 2004), in Morrey spaces over metric measure of non homogenenous type (Sihwaningrum, 2016) and in generalized Morrey spaces over metric measure spaces of non homogeneous type (Sihwaningrum, et al, 2015). This kind of weak type is also established in (Sihwaningrum and Sawano, 2013) for another version of fractional integral operator.…”

A WEAK-(p, q) INEQUALITY FOR FRACTIONAL INTEGRAL OPERATOR ON MORREY SPACES VIA HEDBERG TYPE INEQUALITY

“…Some results on this operator can be found for example in (Adams, 1975), (Nakai, 1994), (Kurata, et al, 2002) and (Eridani, et al, 2014). Other results on the different version of fractional integral operator can be viewed in (Sawano and Shimamura, 2013) and (Sihwaningrum and Sawano, 2013). Meanwhile, the fractional integral operator (equation 1) has been proved to satisfy the weak-(1, q) inequality (where 1 ≤ q < ∞) in Lebesgue spaces (García-Cuerva and Gatto, 2004), in Morrey spaces (Sihwaningrum, 2016a) and in generalized Morrey spaces (Sihwaningrum, et al, 2015).…”

“…Other results on the different version of fractional integral operator can be viewed in (Sawano and Shimamura, 2013) and (Sihwaningrum and Sawano, 2013). Meanwhile, the fractional integral operator (equation 1) has been proved to satisfy the weak-(1, q) inequality (where 1 ≤ q < ∞) in Lebesgue spaces (García-Cuerva and Gatto, 2004), in Morrey spaces (Sihwaningrum, 2016a) and in generalized Morrey spaces (Sihwaningrum, et al, 2015). This fractional integral operator also satisfies the weak-(p, q) inequality (where 1 ≤ p < ∞ and 1 ≤ q < ∞) in Morrey spaces (Sihwaningrum, 2016b).…”

Boundedness of multilinear fractional integrals on metric measure spaces

“…为此, 首先介 绍一般度量测度空间上 Morrey 空间和弱 Morrey 空间的定义. 定义 1.1 [12] [13,14]. 然而, 在一般度量测度空间 (X, d, µ) [15] 获得了在经典 Euclid 空间上多线性分数次积分算子在乘积 Morrey 空间上 的有界性以及当某些或全部 q i 等于 1 时映乘积 Morrey 到弱 Morrey 空间上的有界性.…”