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 ∈ Lloc(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)