Generalized Morrey Spaces Over Nonhomogeneous Metric Measure Spaces

Abstract

Let $({\mathcal{X}},d,\unicode[STIX]{x1D707})$ be a nonhomogeneous metric measure space satisfying the so-called upper doubling and the geometric doubling conditions. In this paper, the authors give the natural definition of the generalized Morrey spaces on $({\mathcal{X}},d,\unicode[STIX]{x1D707})$, and then investigate some properties of the maximal operator, the fractional integral operator and its commutator, and the Marcinkiewicz integral operator.

Weighted Boundedness of Certain Sublinear Operators in Generalized Morrey Spaces on Quasi-Metric Measure Spaces Under the Growth Condition

“…Singular-type operators under more general growth condition (in the sense of [14] and [15]) were studied in [21] and [40]. In [21] the operators T were studied in the nonweighted case, while in [40] they were considered in the weighted space L p,k (X , w) of specific form, which goes back to [20], namely in the case…”

Parameter θ -Type Marcinkiewicz Integral on Nonhomogeneous Weighted Generalized Morrey Spaces

“…For the sake of convenience, the new space is now called a nonhomogeneous metric measure space. Since then, the research on the space has been widely focused, for example, some authors established the properties of function spaces on the nonhomogeneous metric measure space (see [10][11][12][13][14]). On the other hand, the boundedness of singular integral operators on various of spaces is also obtained; the readers can see [15][16][17][18][19][20] and so on.…”

“…Recently, the Morrey space, weighted Morrey space, and generalized Morrey space on ℝ n have been extended to nonhomogeneous metric measure space, for example, we can see [10,13,14]. Motivated by these, in this paper, we first give out the definition of weighted generalized Morrey space and weighted weak generalized Morrey space on ðX, d, μÞ.…”