Weighted and Unweighted Solyanik Estimates for the Multilinear Strong Maximal Function
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Results in Mathematics
Weighted and Unweighted Solyanik Estimates for the Multilinear Strong Maximal Function Moyan Qin and Qingying Xue Abstract. Let ω be a weight in A∗∞ and let Mm n (f ) be the multilinear strong maximal function of f = (f1 , . . . , fm ), where f1 , . . . , fm are functions on Rn . In this paper, we consider the asymptotic estimates for the − distribution functions of Mm n . We show that, for λ ∈ (0, 1), if λ → 1 , m then the multilinear Tauberian constant Cn and the weighted Tauberian m associated with Mm constant Cn,ω n enjoy the properties that 1
Cnm (λ) − 1 m (1 − λ) n
and
m Cn,ω (λ) − 1 m(1 − λ)
4n[ω]A∗
∞
−1
.
Mathematics Subject Classification. Primary 42B20, Secondary 42B35. Keywords. Solyanik estimates, Tauberian constant, multilinear strong maximal function.
1. Introduction It was well known that the Hardy–Littlewood maximal function taking supremum over cubes is of weak type (1, 1) and Lp bounded for p > 1. However, if the supremum is taken over some other kinds of bases, it may enjoy different properties. For example, if the base is composed of all rectangles in Rn with Q. Xue was supported partly by NSFC (Nos. 11671039, 11871101) and NSFC-DFG (No. 11761131002). 0123456789().: V,-vol
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M. Qin and Q. Xue
Results Math
sides parallel to the coordinate axes, then it coincides with the strong maximal function defined by 1 Mn f (x) := sup |f (y)| dy. Rx |R| R Jessen et al. [1] surprisingly pointed out that, unlike the classical HardyLittlewood maximal function, the strong maximal function Mn is not of weak type (1, 1). However, it still enjoys L log L weak type estimate as follows: n−1 |f (x)| + |f (x)| n 1 + log |{x ∈ R : Mn f (x) > λ}| dx, λ λ Rn where log+ t := max (log t, 0). This certain L log L weak type estimate was proved again by C´ ordoba and Fefferman [2], using an alternative geometric method. Since then many nice works have been done for the strong maximal function, we refer the readers to C´ ordoba and Fefferman [9], Bagby [10], Fava et al. [11], Stein [12], Fefferman [13], Cao et al. [14] and the references therein. What we are interested in is the asymptotic estimates for the distribution functions of the maximal functions Mn . In order to state some known results, we need to introduce the Tauberian constant and the weighted Tauberian constant associated with Mn , which are defined by Cn (λ) :=
sup
|{x ∈ Rn : Mn χE (x) > λ}| , |E|
sup
ω ({x ∈ Rn : Mn χE (x) > λ}) , ω (E)
E⊂Rn 0 λ}) . ω (∩m i=1 Ei )
E1 ,...,Em ⊂Rn 0 λ. |Ix | |Ix | Hence, K ⊂ ∪x∈K Ix .
K is a compact set, there exists a finite set of open intervals {Ij } =
Since Ij ∪ Ij ⊂ {Ix }x∈K such that (1) K ⊂ ∪Ij ; |A ∩ Ij | |B ∩ Ij | > λ, ∀j; (2) 2
|Ij | (3) Ij and Ij are sets of disjoint open intervals. Let E1 := ∪Ij and E2 := ∪Ij . Since for every Ij , it holds that |A ∩ Ij | |B ∩ Ij | > λ, |Ij | |Ij |
|A ∩ Ij | ≤ 1, |Ij |
|B ∩ Ij | ≤ 1, |Ij |
then it yields that |B ∩ Ij | |A ∩ Ij | > λ, > λ. |Ij | |Ij |
Recalling that Ij and I
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