Criteria of a multi-weight weak type inequality in Orlicz classes for maximal functions defined on homogeneous type spac
- PDF / 600,300 Bytes
- 13 Pages / 476.22 x 680.315 pts Page_size
- 65 Downloads / 160 Views
DOI: 0
CRITERIA OF A MULTI-WEIGHT WEAK TYPE INEQUALITY IN ORLICZ CLASSES FOR MAXIMAL FUNCTIONS DEFINED ON HOMOGENEOUS TYPE SPACES S. DING and Y. REN∗ School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471000, China e-mails: [email protected], [email protected] (Received January 10, 2020; revised March 1, 2020; accepted March 2, 2020)
Abstract. We obtain some new necessary and sufficient conditions for a multi-weight weak type maximal inequality of the form ϕ(λω1 (x))ω2 (x) dµ ≤ c ϕ(cf (x)ω3 (x))ω4 (x) dµ {x:Mf (x)>λ}
X
in Orlicz classes, where Mf is a Hardy–Littlewood maximal function defined on homogeneous type spaces. Our main result extends some known results.
1. Introduction Weighted theory is one of important parts and the research focuses in harmonic analysis (see [2,4,5,10–18,20,23,25]). A fundamental result in weighted theory stems from Muckenhoupt’s work [19]: let 1 < p < ∞, then the Hardy–Littlewood maximal function is of weak type (p, p) with respect to a pair of weights (u, v) if and only if (u, v) ∈ Ap . Muckenhoupt’s result reveals the deep connection between the boundedness of the Hardy–Littlewood maximal operator in different function spaces and the weight functions. The boundedness of operators in a function space is one of eternal subjects in function space theory. Since the publication of Muckenhoupt’s result, it has become a trend to study the extension of it to various function spaces, which attracts more attention for a long time. In Orlicz classes, there are many excellent related works on this topic, see for example in Gallardo [6], ∗ Corresponding
author. Supported by the National Natural Science Foundation of China (Grant No. 11871195). Key words and phrases: weight, weak type inequality, Hardy–Littlewood maximal function, homogeneous type space, quasi-convex function. Mathematics Subject Classification: 42B25, 46E30. c 2020 0236-5294/$ 20.00 © 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary
22
S. S. DING DING and and Y. Y. REN REN
Bagby [1], Pick [21], Qinsheng [22], Bloom and Kerman [3], Gogatishvili and Kokilashvili [8,9]. Our main aim is to study criteria of a multi-weight weak type maximal inequality of the form ϕ(λω1 (x))ω2(x) dµ ≤ c ϕ(cf (x)ω3(x))ω4(x) dµ (1.1) {x:Mf (x)>λ}
X
in Orlicz classes, where Mf is a Hardy–Littlewood maximal function defined on homogeneous type spaces. This inequality has been considered by Gogatishvili and Kokilashvili in [9]. They showed that the weighted inequality (1.1) holds if and only if one of the following conditions holds: (i) There is a constant c1 > 0 such that the inequality ϕ(c1f (x)ω3(x))ω4 (x) dµ ϕ((f )B ω1 (x))ω2 (x) dµ ≤ c1 B
B
holds for any nonnegative µ-measurable function f : X → R1 and for any ball B. (ii) There are positive constants ε and c2 such that we have the inequality B ϕ(λω1 (x))ω2 (x) dµ ϕ˜ ε ϕ(λω1 (x))ω2 (x) dµ (1.2) ω4 (x) dµ ≤ c2 λµBω3 (x)ω4 (x) B B for any λ > 0 and an arbitrary ball B. One can see t
Data Loading...
