Fractional type Marcinkiewicz integrals over non-homogeneous metric measure spaces

Journal of Inequalities and Applications, Oct 2016

The main goal of the paper is to establish the boundedness of the fractional type Marcinkiewicz integral M β , ρ , q on non-homogeneous metric measure space which includes the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel satisfies a certain Hörmander-type condition, the authors prove that M β , ρ , q is bounded from Lebesgue space L 1 ( μ ) into the weak Lebesgue space L 1 , ∞ ( μ ) , from the Lebesgue space L ∞ ( μ ) into the space RBLO ( μ ) , and from the atomic Hardy space H 1 ( μ ) into the Lebesgue space L 1 ( μ ) . Moreover, the authors also get a corollary, that is, M β , ρ , q is bounded on L p ( μ ) with 1 < p < ∞ . MSC: non-homogeneous metric measure space, fractional type Marcinkiewicz integral, Lebesgue space, Hardy space, RBLO(μ)$\operatorname{RBLO}(\mu)$.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

http://www.journalofinequalitiesandapplications.com/content/pdf/s13660-016-1203-0.pdf

Fractional type Marcinkiewicz integrals over non-homogeneous metric measure spaces

Lu and Tao Journal of Inequalities and Applications Fractional type Marcinkiewicz integrals over non-homogeneous metric measure spaces Guanghui Lu Shuangping Tao The main goal of the paper is to establish the boundedness of the fractional type Marcinkiewicz integral Mβ,ρ,q on non-homogeneous metric measure space which includes the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel satisfies a certain Hörmander-type condition, the authors prove that Mβ,ρ,q is bounded from Lebesgue space L1(μ) into the weak Lebesgue space L1,∞(μ), from the Lebesgue space L∞(μ) into the space RBLO(μ), and from the atomic Hardy space H1(μ) into the Lebesgue space L1(μ). Moreover, the authors also get a corollary, that is, Mβ,ρ,q is bounded on Lp(μ) with 1 < p < ∞. MSC: non-homogeneous metric measure space; fractional type Marcinkiewicz integral; Lebesgue space; Hardy space; RBLO(μ) 1 Introduction In , Hytönen in [] first introduced a new class of metric measure spaces which satisfy the so-called upper doubling and the geometrically doubling conditions (see also Definitions . and . below, respectively), for convenience, the new spaces are called nonhomogeneous metric measure spaces. As special cases, the new spaces not only contain the homogeneous type spaces (see []), but also they include metric spaces endowed with measures satisfying the polynomial growth condition (see, for example, [–]). Further, it is meaningful to pay much attention to a study of the properties of some classical operators, commutators, and function spaces on non-homogeneous metric measure spaces; see [–]. In addition, we know that the harmonic analysis has important applications in many fields including geometrical analysis, functional analysis, partial differential equations, and fuzzy fractional differential equations, we refer the reader to [–] and the sense of Hytönen []. In , Hu et al. [] obtained the boundedness of the Marcinkiewicz with non-doubling measure. Besides, Lin and Yang [] established some equivalent sider the boundedness of the fractional type Marcinkiewicz integrals introduced in [] © 2016 Lu and Tao. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. To state the main consequences of this article, we first of all recall some necessary notions and notation. Hytönen [] originally introduced the following notions of the upper doubling condition and the geometrically doubling condition. Definition . ([]) A metric measure space (X , d, μ) is said to be upper doubling if μ is a Borel measure on X and there exist a dominating function λ : X × (, ∞) → (, ∞) and a positive constant Cλ such that, for each x ∈ X , r → λ(x, r) is non-decreasing and, for all x ∈ X and r ∈ (, ∞), μ B(x, r) ≤ λ(x, r) ≤ Cλλ x, r . where x, y ∈ X and d(x, y) ≤ r. Based on this, we also assume the dominating function λ that in (.) satisfies (.) in this paper. Definition . ([]) A metric space (X , d) is said to be geometrically doubling, if there exists some N ∈ N such that, for any ball B(x, r) ⊂ X , there exists a finite ball covering {B(xi, r )}i of B(x, r) such that the cardinality of this covering is at most N. KB,S :=  + where cB is the center of the ball B. Though the measure doubling condition is not assumed uniformly for all balls on (X , d, μ), it was proved in [] that there still exist many balls satisfying the property of the (α, η)-doubling, namely, we say that a ball B ⊂ X is (α, η)-doubling if μ(αB) ≤ ημ(B), for α, η > . In the rest of this paper, unless α and ηα are specified, otherwise, by an (α, ηα)doubling ball we mean a (, β)-doubling ball with a fixed number η > max{Cλ log , n}, where n := log N is viewed as a geometric dimension of the space. Moreover, the smallest |fB – fR| ≤ CKB,R ess inf f – ess inf f ≤ CKB,S B S for any two balls B and R such that B ⊂ R. Moreover, the RBMO(μ) norm of f is defined to be the minimal constant C as above and denoted by f RBMO(μ). From [], Hytönen showed that the space RBMO(μ) is not dependent on the choice of κ . Lin and Yang [] introduced the following definition of the space RBLO(μ) and proved that RBLO(μ) ⊂ RBMO(μ). Definition . ([]) A function f ∈ Lloc(μ) is said to belong to the space RBLO(μ) if there exists a positive constant C such that. for any (, β)-doubling ball B, for any two (, β)-doubling balls B ⊂ S. The minimal constant C above is defined to be the norm of f in RBLO(μ) and denoted by f RBLO(μ). [d(x, y)]+β K (x, y) ≤ C λ(x, d(x, y)) , and for all x, x˜, y ∈ X with d(x, y) ≥ d(x, x˜), [d(x, x˜)]δ++β K (x, y) – K (x˜, y) + K (y, x) – K (...truncated)


This is a preview of a remote PDF: http://www.journalofinequalitiesandapplications.com/content/pdf/s13660-016-1203-0.pdf

Guanghui Lu, Shuangping Tao. Fractional type Marcinkiewicz integrals over non-homogeneous metric measure spaces, Journal of Inequalities and Applications, 2016, pp. 259, 2016, DOI: 10.1186/s13660-016-1203-0