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Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 923874, 21 pages
http://dx.doi.org/10.1155/2012/923874
Research Article

BMO-Boundedness of Maximal Operators and g-Functions Associated with Laguerre Expansions

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China

Received 29 February 2012; Revised 21 May 2012; Accepted 21 June 2012

Academic Editor: AurelianΒ Gheondea

Copyright Β© 2012 Li Cha and Heping Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let {πœ‘π›Όπ‘›}π‘›βˆˆβ„• be the Laguerre functions of Hermite type with index 𝛼. These are eigenfunctions of the Laguerre differential operator 𝐿𝛼=1/2(βˆ’π‘‘2/𝑑𝑦2+𝑦2+ξ€·1/𝑦2ξ€Έ(𝛼2βˆ’1/4)). In this paper, we investigate the boundedness of the Hardy-Littlewood maximal function, the heat maximal function, and the Littlewood-Paley 𝑔-function associated with 𝐿𝛼 in the localized BMO space BMO𝐿𝛼, which is the dual space of the Hardy space 𝐻1𝐿𝛼.

1. Introduction

Let π‘›βˆˆβ„•, 𝛼>βˆ’1. The Laguerre function of Hermite type πœ‘π›Ό on (0,∞) is defined as πœ‘π›Όπ‘›ξ‚΅(𝑦)=Ξ“(𝑛+1)ξ‚ΆΞ“(𝑛+1+𝛼)1/2π‘’βˆ’π‘¦2/2𝑦𝛼𝐿𝛼𝑛𝑦2ξ€Έ(2𝑦)1/2,π‘¦βˆˆ(0,∞),(1.1) where 𝐿𝛼𝑛(π‘₯) denotes the Laguerre polynomial of degree 𝑛 and order 𝛼, see [1]. It is well known that for every 𝛼>βˆ’1 the system {πœ‘π›Όπ‘›}βˆžπ‘›=0 forms an orthonormal basis of 𝐿2(0,∞). Moreover, these functions are eigenfunctions of the Laguerre differential operator 𝐿𝛼=12ξ‚΅βˆ’π‘‘2𝑑𝑦2+𝑦2+1𝑦2𝛼2βˆ’14(1.2) satisfying πΏπ›Όπœ‘π›Όπ‘›=(2𝑛+𝛼+1)πœ‘π›Όπ‘›. The operator 𝐿𝛼 can be extended to a positive self-adjoint operator on 𝐿2(0,∞) by giving a suitable domain of definition, see [2]; we also denote the extension by 𝐿𝛼. Let {𝑇𝛼𝑑}𝑑β‰₯0 be the heat-diffusion semigroup generated by βˆ’πΏπ›Ό. More precisely, for π‘“βˆˆπΏ2(0,∞), we define π‘‡π›Όπ‘‘ξ€œπ‘“(π‘₯)=∞0π‘Šπ›Όπ‘‘(π‘₯,𝑦)𝑓(𝑦)𝑑𝑦,(1.3) where π‘Šπ›Όπ‘‘ξ‚΅(π‘₯,𝑦)=2π‘’βˆ’π‘‘1βˆ’π‘’βˆ’2𝑑1/2ξ‚΅2π‘₯π‘¦π‘’βˆ’π‘‘1βˆ’π‘’βˆ’2𝑑1/2𝐼𝛼2π‘₯π‘¦π‘’βˆ’π‘‘1βˆ’π‘’βˆ’2π‘‘ξ‚Άξ‚΅βˆ’1exp21+π‘’βˆ’2𝑑1βˆ’π‘’βˆ’2𝑑π‘₯2+𝑦2ξ€Έξ‚Ά.(1.4)𝐼𝛼 is the modified Bessel function of the first kind and order 𝛼.

In [3], we introduced and developed a localized BMO space BMO𝐿𝛼 associated with the operator 𝐿𝛼, which is the dual space of the Hardy space 𝐻1𝐿𝛼 introduced by DziubaΕ„ski [4]. More precisely, let πœŒπΏπ›Ό1(π‘₯)=8ξ‚€1minπ‘₯,π‘₯,π‘₯>0.(1.5)

Definition 1.1. Let 𝛼>βˆ’1/2, 𝐡𝑠(𝑦) be any ball in (0,∞) with the center 𝑦 and the radius 𝑠 and 𝑓 a locally integrable function on (0,∞). We say π‘“βˆˆBMO𝐿𝛼 if there exists a constant 𝐢β‰₯0 independent of 𝑠 and 𝑦 such that 1||𝐡𝑠||ξ€œ(𝑦)𝐡𝑠(𝑦)||π‘“βˆ’π‘“π΅π‘ (𝑦)||≀𝐢,if𝑠<πœŒπΏπ›Ό(1𝑦),||𝐡𝑠||ξ€œ(𝑦)𝐡𝑠(𝑦)||𝑓||≀𝐢,if𝑠β‰₯πœŒπΏπ›Ό(𝑦).(1.6) Here, 𝑓𝐡𝑠(𝑦)=(1/|π΅π‘ βˆ«(𝑦)|)𝐡𝑠(𝑦)𝑓𝑑π‘₯. We let ‖𝑓‖BMO𝐿𝛼 denote the smallest 𝐢 in the two inequalities above.

It is readily seen that BMO𝐿𝛼 is a Banach space with norm β€–β‹…β€–BMO𝐿𝛼.

In this paper, we obtain the boundedness on BMO𝐿𝛼 of several operators including the Hardy-Littlewood maximal operator defined on (0,∞), the heat maximal function, and the Littlewood-Paley 𝑔-function associated with 𝑇𝛼𝑑.

These results were investigated by DziubaΕ„ski et al. in [5] for SchrΓΆdinger operators on ℝ𝑑 with 𝑑β‰₯3 and with potentials satisfying a reverse HΓΆlder's inequality. Recently, a theory of localized BMO spaces on RD-spaces associated with an admissible function 𝜌 was investigated in [6]; the authors also established the similar results above for their BMO spaces. The admissible function 𝜌 in [6] is required to satisfy 1𝜌(π‘₯)≀𝐢01ξ‚΅πœŒ(𝑦)1+𝑑(π‘₯,𝑦)ξ‚ΆπœŒ(𝑦)π‘˜0.(1.7) Obviously, our πœŒπΏπ›Ό in (1.5) does not satisfy this condition. Indeed, let π‘₯ tend to zero and 𝑦=1; then the left side becomes greater than the right.

It is notable the generalized square functions associated to SchrΓΆdinger operators are studied in [7]. The authors of [7] gave several of equivalent conditions for BMO-boundedness of square functions.

In this paper, in order to obtain some key estimates, we will employ the differences in integral kernels (the heat kernel, the 𝑔-function kernel) associated with the Hermite operator and the Laguerre operator, respectively (see [8, 9]).

The paper is organized as follows. In the next section we present some preliminary lemmas and collect some useful estimates of the kernels associated with the heat semigroups and the 𝑔-functions. In Section 3, we establish the boundedness of two maximal operators (the Hardy-Littlewood maximal operator and the heat maximal function) from BMO𝐿𝛼 to BMO𝐿𝛼. In Section 4, we obtain the boundedness on BMO𝐿𝛼 of the Littlewood-Paley 𝑔-function associated with the heat semigroup for 𝐿𝛼. We make some conventions. Throughout this paper by 𝐢 we always denote a positive constant that may vary at each occurrence; π΅π‘Ÿ(𝑦0) stands for {𝑦>0,|π‘¦βˆ’π‘¦0|β‰€π‘Ÿ}; 𝐴∼𝐡 means (1/𝐢)𝐴≀𝐡≀𝐢𝐴, and the notation π‘‹β‰²π‘Œ is used to indicate that π‘‹β‰€πΆπ‘Œ with an independent positive constant 𝐢.

2. Preliminaries

Now we give the following covering lemma for (0,∞) which will be used frequently below. The proof is trivial and left to the reader.

Lemma 2.1. Let π‘₯0=1,   π‘₯𝑗=π‘₯π‘—βˆ’1+πœŒπΏπ›Ό(π‘₯π‘—βˆ’1) for 𝑗β‰₯1, and π‘₯𝑗=π‘₯𝑗+1βˆ’πœŒπΏπ›Ό(π‘₯𝑗+1) for 𝑗<0. One defines the family of β€œcritical balls” of ℬ={π΅π‘˜}βˆžπ‘˜=βˆ’βˆž, where π΅π‘˜βˆΆ={π‘₯∈(0,∞)∢|π‘₯βˆ’π‘₯π‘˜|<πœŒπΏπ›Ό(π‘₯π‘˜)}. Then one has(a)β‹ƒβˆžπ‘˜=βˆ’βˆžπ΅π‘˜=(0,∞), (b)for every π‘˜βˆˆβ„€, π΅π‘˜βˆ©π΅π‘—=βˆ… provided that π‘—βˆ‰{π‘˜βˆ’1,π‘˜,π‘˜+1}, (c)for any 𝑦0∈(0,∞), at most three balls in ℬ have nonempty intersection with 𝐡(𝑦0,πœŒπΏπ›Ό(𝑦0)).

Corollary 2.2. There exists a constant 𝐢>0 such that for every 𝐡𝑅(π‘₯)βŠ†(0,∞) with 𝑅>πœŒπΏπ›Ό(π‘₯), one has ||𝐡𝑅(||≀π‘₯)ξ€½π΅π‘˜βˆˆβ„¬βˆΆπ΅π‘˜βˆ©π΅π‘…ξ€Ύ(π‘₯)β‰ βˆ…||π΅π‘˜||||𝐡≀𝐢𝑅(||.π‘₯)(2.1)

Corollary 2.3. There exists a constant 𝐢 such that, for π‘“βˆˆBMO𝐿𝛼, one has ‖𝑓‖BMO𝐿𝛼≀𝐢supπ‘˜ξ‚€||𝑓||π΅π‘˜+‖𝑓‖BMO(π΅βˆ—π‘˜),(2.2) where, for any ball 𝐡, the norm β€–β‹…β€–BMO(𝐡) is given by ‖𝑓‖BMO(𝐡)=supπ΅π‘Ÿ(π‘₯)βŠ‚π΅1π΅π‘Ÿξ€œ(π‘₯)π΅π‘Ÿ(π‘₯)||π‘“βˆ’π‘“π΅π‘Ÿ(π‘₯)||π‘‘π‘¦βˆΌsupπ΅π‘Ÿ(π‘₯)βŠ‚π΅infπ‘βˆˆβ„‚1π΅π‘Ÿξ€œ(π‘₯)π΅π‘Ÿ(π‘₯)||||π‘“βˆ’π‘π‘‘π‘¦.(2.3)

Corollary 2.4 (see [3, Corollary  3]). Let 𝐡=π΅π‘Ÿ(𝑦0)βŠ‚(0,∞). There exists a constant 𝐢>0 such that, for all π‘“βˆˆBMO𝐿𝛼, one has (1)if π‘Ÿβ‰₯πœŒπΏπ›Ό(𝑦0)/2, then ∫((1/|𝐡|)𝐡|𝑓(π‘₯)|2𝑑π‘₯)1/2≀𝐢‖𝑓‖BMO𝐿𝛼, (2)if π‘Ÿ<πœŒπΏπ›Ό(𝑦0)/2, then ∫((1/|𝐡|)𝐡|𝑓(π‘₯)βˆ’π‘“π΅|2𝑑π‘₯)1/2≀𝐢‖𝑓‖BMO𝐿𝛼. We give two elementary lemmas, which will be used frequently in next section. The proofs are trivial, and the reader also refer to Lemmas 9 and 2 in [5].

Lemma 2.5. Let β„ŽβˆˆBMO(π΅βˆ—π‘˜) and 𝑔1 and 𝑔2 be functions in 𝐿∞(0,∞). If 𝑓 is any measurable function satisfying β„Ž+𝑔1β‰€π‘“β‰€β„Ž+𝑔2,a.e.,(2.4) then π‘“βˆˆBMO(π΅βˆ—π‘˜) and ‖𝑓‖BMO(π΅βˆ—π‘˜)β‰€β€–β„Žβ€–BMO(π΅βˆ—π‘˜)+max(‖𝑔1β€–βˆž,‖𝑔2β€–βˆž).

Lemma 2.6. For all π‘“βˆˆBMO𝐿𝛼 and 𝐡=π΅π‘Ÿ(𝑦0) with π‘Ÿ<πœŒπΏπ›Ό(𝑦0). There exists a constant 𝐢>0 such that ||π‘“π΅βˆ—||ξƒ©πœŒβ‰€πΆ1+log𝐿𝛼𝑦0ξ€Έπ‘Ÿξƒͺ‖𝑓‖BMO𝐿𝛼.(2.5)

Let 𝐻 be the Hermite operator 1𝐻=2ξ‚΅βˆ’π‘‘2𝑑π‘₯2+π‘₯2ξ‚Ά.(2.6) One considers the heat diffusion semigroup {π‘Šπ‘‘}𝑑>0 associated with 𝐻 and defined by, for every π‘“βˆˆπΏ2(ℝ), π‘Šπ‘‘ξ€œπ‘“(π‘₯)=β„π‘Šπ‘‘(π‘₯,𝑦)𝑓(𝑦)𝑑𝑦,π‘₯βˆˆβ„,(2.7) where for each π‘₯,π‘¦βˆˆβ„ and 𝑑>0, π‘Šπ‘‘ξƒ¬π‘’(π‘₯,𝑦)=βˆ’π‘‘πœ‹ξ€·1βˆ’π‘’βˆ’2𝑑1/2ξ‚΅βˆ’1exp21+π‘’βˆ’2𝑑1βˆ’π‘’βˆ’2𝑑π‘₯2+𝑦2𝑒+2π‘₯π‘¦βˆ’π‘‘1βˆ’π‘’βˆ’2𝑑(2.8) (see [10]).

Proposition 2.7. Let 𝛼>βˆ’1/2, π‘Šπ‘‘(π‘₯,𝑦) be in (2.8). There exists 𝐢>0 such that, for 𝑑>0, (a)π‘Šπ›Όπ‘‘(π‘₯,𝑦)≀𝐢𝑦𝛼+1/2π‘₯βˆ’π›Όβˆ’3/2, 0<𝑦<π‘₯/2, (b)π‘Šπ›Όπ‘‘(π‘₯,𝑦)≀𝐢π‘₯𝛼+1/2π‘¦βˆ’π›Όβˆ’3/2, 0<2π‘₯<𝑦, (c)|π‘Šπ›Όπ‘‘(π‘₯,𝑦)βˆ’π‘Šπ‘‘(π‘₯,𝑦)|≀𝐢(1/𝑦), π‘₯/2<𝑦<2π‘₯, (d)|π‘Šπ›Όπ‘‘βˆš(π‘₯,𝑦)|≀𝐢(1/𝑑)π‘’βˆ’|π‘₯βˆ’π‘¦|2/10𝑑.

Parts (a), (b), and (c) are the contents of Lemma 2.11 in [8]. Part (d) is from (2.6) in [4].

Remark 2.8. The ranges 0<𝑦<π‘₯/2 and 0<2π‘₯<𝑦 are not critical; Proposition 2.7 also holds when 0<𝑦<π‘₯/𝑐 and 0<𝑐π‘₯<𝑦, where 𝑐>1.

Now we consider the estimates of the integral kernel for the 𝑔-function, which will be defined in Section 4: 𝑄𝑑(π‘₯,𝑦)=𝑑2πœ•π‘Šπ›Όπ‘ (π‘₯,𝑦)||||πœ•π‘ π‘ =𝑑2,(2.9)𝑃𝑑(π‘₯,𝑦)=𝑑2πœ•π‘Šπ‘ (π‘₯,𝑦)||||πœ•π‘ π‘ =𝑑2.(2.10)

Proposition 2.9. One has,(a)for every 𝑑,π‘₯,π‘¦βˆˆ(0,∞) such that π‘’βˆ’π‘‘2π‘₯𝑦/(1βˆ’π‘’βˆ’2𝑑2)≀1, ||𝑄𝑑(||π‘₯,𝑦)≀𝐢𝑑2(π‘₯𝑦)(𝛼+1/2)π‘’βˆ’(π‘₯2+𝑦2)/8𝑑2π‘’βˆ’(𝛼+1)𝑑2ξ€·1βˆ’π‘’βˆ’2𝑑2𝛼+2,(2.11)(b)for every 𝑑,π‘₯,π‘¦βˆˆ(0,∞) such that π‘’βˆ’π‘‘2π‘₯𝑦/(1βˆ’π‘’βˆ’2𝑑2)>1, ||𝑄𝑑(π‘₯,𝑦)βˆ’π‘ƒπ‘‘(||π‘₯,𝑦)≀𝐢𝑑2π‘’βˆ’(π‘₯βˆ’π‘¦)2/2𝑑2𝑒𝑑2/2ξ€·π‘₯𝑦1βˆ’π‘’βˆ’π‘‘2ξ€Έ1/2.(2.12)

Parts (a) and (b) are contained in [9, (3.4) and (3.6)].

Proposition 2.10. For every 𝑁β‰₯1, there is a constant 𝐢𝑁 such that (a)if 𝑑>0, |𝑃𝑑(π‘₯,𝑦)|β‰€πΆπ‘π‘’βˆ’π‘‘2/8(1/𝑑)π‘’βˆ’|π‘₯βˆ’π‘¦|2/10𝑑2(1+𝑑|π‘₯|)βˆ’π‘; (b)for |β„Ž|≀𝑑, |𝑃𝑑(π‘₯+β„Ž,𝑦)βˆ’π‘ƒπ‘‘(π‘₯,𝑦)|≀𝐢(|β„Ž|/𝑑)(1/𝑑)π‘’βˆ’|π‘₯βˆ’π‘¦|2/20𝑑2, 𝐢 is independent of π‘₯,𝑦,𝑑; (c)|βˆ«βˆžβˆ’βˆžπ‘ƒπ‘‘(π‘₯,𝑦)𝑑𝑦|≀𝐢(𝑑/πœŒπΏπ›Ό(π‘₯))2, 𝐢 is independent of π‘₯ and 𝑑.

Proof. By using (2.8) we can write, for every π‘₯,π‘¦βˆˆβ„ and 𝑠>0, πœ•π‘Šπœ•π‘ π‘ 1(π‘₯,𝑦)=βˆ’2βˆšπœ‹π‘’βˆ’((π‘₯βˆ’π‘’βˆ’π‘ π‘¦)2+(π‘¦βˆ’π‘’βˆ’π‘ π‘₯)2)/2(1βˆ’π‘’βˆ’2𝑠)π‘’βˆ’π‘ /2ξ€·1βˆ’π‘’βˆ’2𝑠3/2Γ—ξ‚»1+π‘’βˆ’2𝑠+2π‘’βˆ’π‘ (𝑦(π‘₯βˆ’π‘’βˆ’π‘ π‘¦)+π‘₯(π‘¦βˆ’π‘’βˆ’π‘ π‘₯))βˆ’2π‘’βˆ’2𝑠(π‘₯βˆ’π‘’βˆ’π‘ π‘¦)2+(π‘¦βˆ’π‘’βˆ’π‘ π‘₯)21βˆ’π‘’βˆ’2𝑠.(2.13)
By the simple fact (π‘₯βˆ’π‘’βˆ’π‘ π‘¦)2+(π‘¦βˆ’π‘’βˆ’π‘ π‘₯)2=2(π‘₯βˆ’π‘¦)2π‘’βˆ’π‘ +ξ€·π‘₯2+𝑦2ξ€Έ(1βˆ’π‘’βˆ’π‘ )2,(2.14) a straightforward manipulation leads to |||πœ•π‘Šπœ•π‘ π‘ |||(π‘₯,𝑦)β‰€πΆπ‘’βˆ’((π‘₯βˆ’π‘¦)2+(π‘₯2+𝑦2)(1βˆ’π‘’βˆ’π‘ )2)/8(1βˆ’π‘’βˆ’2𝑠)π‘’βˆ’π‘ /2ξ€·1βˆ’π‘’βˆ’2𝑠3/2β‰€πΆπ‘π‘’βˆ’π‘ /8ξ€·1βˆ’π‘’βˆ’2𝑠3/2π‘’βˆ’π‘(|π‘₯βˆ’π‘¦|2/10𝑠)ξ‚€βˆš1+𝑠|π‘₯|βˆ’π‘,(2.15) which implies (a).
To prove (b), we also directly compute the π‘₯ partial derivative: πœ•π‘Šπœ•π‘₯πœ•π‘ π‘ 1(π‘₯,𝑦)=βˆ’2βˆšπœ‹π‘’βˆ’((π‘₯βˆ’π‘’βˆ’π‘ π‘¦)2+(π‘¦βˆ’π‘’βˆ’π‘ π‘₯)2)/2(1βˆ’π‘’βˆ’2𝑠)π‘’βˆ’π‘ /2ξ€·1βˆ’π‘’βˆ’2𝑠3/2Γ—ξ‚»4π‘’βˆ’π‘ (π‘¦βˆ’π‘’βˆ’π‘ π‘₯)βˆ’4π‘’βˆ’2𝑠(π‘₯βˆ’π‘¦)1+π‘’βˆ’π‘ ξ‚Ό+12βˆšπœ‹π‘’βˆ’((π‘₯βˆ’π‘’βˆ’π‘ π‘¦)2+(π‘¦βˆ’π‘’βˆ’π‘ π‘₯)2)/2(1βˆ’π‘’βˆ’2𝑠)π‘’βˆ’π‘ /2ξ€·1βˆ’π‘’βˆ’2𝑠5/2((π‘₯βˆ’π‘’βˆ’π‘ π‘¦)βˆ’π‘’βˆ’π‘ (π‘¦βˆ’π‘’βˆ’π‘ π‘₯Γ—ξ‚»))1+π‘’βˆ’2𝑠+2π‘’βˆ’π‘ (𝑦(π‘₯βˆ’π‘’βˆ’π‘ π‘¦)+π‘₯(π‘¦βˆ’π‘’βˆ’π‘ π‘₯))βˆ’2π‘’βˆ’2𝑠(π‘₯βˆ’π‘’βˆ’π‘ π‘¦)2+(π‘¦βˆ’π‘’βˆ’π‘ π‘₯)21βˆ’π‘’βˆ’2𝑠.(2.16) By an elementary manipulation and (2.14), we have |||πœ•π‘Šπœ•π‘₯πœ•π‘ π‘ |||1(π‘₯,𝑦)≀𝐢𝑠2π‘’βˆ’(π‘₯βˆ’π‘¦)2/16𝑠.(2.17) This together with the mean value theorem and the condition |β„Ž|≀𝑑 leads to (b).
Let πœ™π‘›(𝑦)=πœ™(𝑦/𝑛); πœ™(𝑦) is a smooth function satisfying πœ™(𝑦)=1 for |𝑦|≀1, πœ™(𝑦)=0 for |𝑦|β‰₯2 and Ξ”πœ™(𝑦)≀1 for π‘¦βˆˆβ„. From the above, for fixed 𝑠 and π‘₯, a straightforward manipulation shows that ξ€œ+βˆžβˆ’βˆž||||πœ•π‘Šπ‘ (π‘₯,𝑦)||||πœ•π‘ π‘‘π‘¦<∞.(2.18) Hence, we have ||||ξ€œβˆžβˆ’βˆžπœ•π‘Šπ‘ (π‘₯,𝑦)||||=||||πœ•π‘ π‘‘π‘¦limπ‘›β†’βˆžξ€œβˆžβˆ’βˆžπœ•π‘Šπ‘ (π‘₯,𝑦)πœ™πœ•π‘ π‘›||||=||||(𝑦)𝑑𝑦limπ‘›β†’βˆžξ€œβˆžβˆ’βˆžπ‘Šπ‘ (π‘₯,𝑦)π»πœ™π‘›(||||ξ€œπ‘¦)π‘‘π‘¦β‰€πΆβˆžβˆ’βˆžπ‘Šπ‘ (π‘₯,𝑦)𝑦2𝑑𝑦.(2.19) Using (2.8) again, ||||ξ€œπΌ=βˆžβˆ’βˆžπœ•π‘Šπ‘ (π‘₯,𝑦)||||ξ€œπœ•π‘ π‘‘π‘¦β‰€πΆβˆžβˆ’βˆžπ‘’βˆ’π‘ /4βˆšπ‘ ξƒ©βˆ’exp(π‘₯βˆ’π‘¦)2π‘’βˆ’π‘ +ξ€·π‘₯2+𝑦2ξ€Έ(1βˆ’π‘’βˆ’π‘ )22ξ€·1βˆ’π‘’βˆ’2𝑠ξƒͺξ€·(π‘¦βˆ’π‘₯)2+π‘₯2𝑑𝑦,(2.20) which implies (c).

Lemma 2.11 (see [3, Theorem 2]). For all π‘“βˆˆBMO𝐿𝛼 and 𝐡=π΅π‘Ÿ(𝑦0)βŠ†(0,∞), there exists a constant 𝐢>0 such that 1||𝐡||ξ€œπ‘Ÿ0ξ€œπ΅π‘„2𝑑𝑓(π‘₯)𝑑π‘₯𝑑𝑑𝑑≀𝐢‖𝑓‖2BMO𝐿𝛼.(2.21)

3. Maximal Operators

First of all, we define the following notions: 𝑀+𝑓(π‘₯)=supπ‘₯βˆˆπ΅βŠ†(0,∞)1||𝐡||ξ€œπ΅||||ℐ𝑓(𝑦)𝑑𝑦,(3.1)βˆ—π›Όπ‘“(π‘₯)=sup𝑑>0||𝑇𝛼𝑑||.𝑓(π‘₯)(3.2)

In this section, we will show β„βˆ—π›Ό and 𝑀+ are bounded on BMO𝐿𝛼.

Theorem 3.1. There exists a constant 𝐢>0 such that, for all π‘“βˆˆBMO𝐿𝛼, 𝑀+𝑓<∞, for a.e. π‘₯∈(0,∞), and ‖‖𝑀+𝑓‖‖BMO𝐿𝛼≀𝐢‖𝑓‖BMO𝐿𝛼.(3.3)

Proof. First of all, we show that for a.e. π‘₯∈(0,∞), 𝑀+𝑓<∞. To do this, we only need to show that for, at almost π‘₯βˆˆπ΅π‘˜βŠ†β„¬ in Lemma 2.1, 𝑀+𝑓(π‘₯)<∞. Let us split 𝑓=𝑓1+𝑓2 with 𝑓1=π‘“πœ’π΅βˆ—π‘˜. Obviously, since 𝑓 is locally integrable, we have 𝑀+𝑓1<∞ for a.e. π‘₯∈(0,∞). For 𝑓2, if π‘₯∈𝐡 and π΅βˆ©π΅βˆ—π‘˜=βˆ…, since supp 𝑓2 is in the complement of π΅βˆ—π‘˜, we have ∫(1/|𝐡|)𝐡|𝑓(𝑦)|𝑑𝑦=0. Otherwise, by the definition of BMO𝐿𝛼, ∫(1/|𝐡|)𝐡|𝑓(𝑦)|𝑑𝑦≀(4/|𝐡4π‘Ÿ(π‘₯π‘˜βˆ«)|)𝐡4π‘Ÿ(π‘₯π‘˜)|𝑓(𝑦)|𝑑𝑦≀𝑐‖𝑓‖BMO𝐿𝛼.
We turn to the boundedness in BMO𝐿𝛼. Let 𝑀 denote the Hardy-Littlewood function on ℝ; it is well known in [11] that 𝑀 is bounded on BMO(ℝ). Let 𝑓0 be a function defined on ℝ which is 𝑓 on (0,∞) and 0 on (βˆ’βˆž,0]. Notice that 𝑀+𝑓=𝑀𝑓0, for π‘₯∈(0,∞), so ‖‖𝑀+𝑓‖‖BMO(π΅βˆ—π‘˜)=‖‖𝑀𝑓0β€–β€–BMO(π΅βˆ—π‘˜)‖‖𝑓≀𝐢0β€–β€–BMO.(3.4) Now, we need to show that ‖𝑓0β€–BMO≀𝐢‖𝑓‖BMO𝐿𝛼. Indeed, if π΅βŠ†(0,∞), it is obvious that ∫(1/|𝐡|)𝐡|𝑓0βˆ’(𝑓0)𝐡|𝑑𝑦≀‖𝑓‖BMO𝐿𝛼. If 𝐡∩(0,∞)=βˆ…, then ∫(1/|𝐡|)𝐡|𝑓0βˆ’(𝑓0)𝐡|𝑑𝑦=0. If 𝐡∩(0,∞)β‰ βˆ… and 𝐡∩(βˆ’βˆž,0)β‰ βˆ…, let 𝐡=𝐡1βˆͺ𝐡2, here 𝐡1=𝐡∩(βˆ’βˆž,0) and 𝐡2=𝐡∩(0,∞), then 1||𝐡||ξ€œπ΅||𝑓0βˆ’ξ€·π‘“0𝐡||1𝑑𝑦≀2||𝐡||ξ€œπ΅2||𝑓0||𝑑𝑦≀2‖𝑓‖BMO𝐿𝛼.(3.5) On the other hand, we again split 𝑓=𝑓1+𝑓2 with 𝑓1=π‘“πœ’π΅βˆ—π‘˜, from the argument above, 𝑀+𝑓2(π‘₯)≀𝑐‖𝑓‖BMO𝐿𝛼, for a.e π‘₯βˆˆπ΅π‘˜. So 1||π΅π‘˜||ξ€œπ΅π‘˜||𝑀+𝑓||1𝑑𝑦≀||π΅π‘˜||ξ€œπ΅π‘˜||𝑀+𝑓1||1𝑑𝑦+||π΅π‘˜||ξ€œπ΅π‘˜||𝑀+𝑓2||≲1𝑑𝑦||π΅π‘˜||ξ€œπ΅π‘˜||𝑀+𝑓1||2𝑑𝑦1/2+‖𝑓‖BMO𝐿𝛼≲‖𝑓‖BMO𝐿𝛼,(3.6) where in the last inequality we have used Corollary 2.4.

Theorem 3.2. Let 𝛼>βˆ’1/2. There exists a constant 𝐢>0 such that β€–β€–β„βˆ—π›Όπ‘“β€–β€–BMO𝐿𝛼≀𝐢‖𝑓‖BMO𝐿𝛼.(3.7)

Proof. By the definition of BMO𝐿𝛼 and Corollary 2.3, it suffices to prove the following: for every fixed β€œcritical ball” π΅π‘˜βˆˆβ„¬ (see Lemma 2.1) we have (1)∫(1/|𝐡_π‘˜|)π΅π‘˜|β„βˆ—π›Όπ‘“|𝑑π‘₯≀𝐢‖𝑓‖BMO𝐿𝛼, (2)β€–β„βˆ—π›Όπ‘“β€–BMO(π΅βˆ—π‘˜)≀𝐢‖𝑓‖BMOL𝛼.
Let us start to prove (1). It is immediate from Theorem 3.1 and (d) of Proposition 2.7; since β„βˆ—π›Όπ‘“(π‘₯)≀𝑀+𝑓(π‘₯), for π‘₯>0, therefore, 1||π΅π‘˜||ξ€œπ΅π‘˜||𝑀+𝑓||𝑑𝑦≀𝐢‖𝑓‖BMO𝐿𝛼.(3.8)
It remains to show (2). By Lemma 2.5, we split β„βˆ—π›Όπ‘“(π‘₯) into several parts. First, we shall show β€–β€–β€–β€–β€–sup𝑑>𝜌2𝐿𝛼π‘₯π‘˜ξ€Έ||𝑇𝛼𝑑||‖‖‖‖‖𝑓(π‘₯)𝐿∞(π΅βˆ—π‘˜)≀𝐢‖𝑓‖BMO𝐿𝛼.(3.9)
From (d) of Proposition 2.7, we have ||𝑇𝛼𝑑||β‰²ξ€œπ‘“(π‘₯)∞0||||1𝑓(𝑦)𝑑1/2||||1+π‘₯βˆ’π‘¦βˆšπ‘‘ξƒͺβˆ’π‘β‰²π‘‘π‘¦βˆžξ“π‘—=012𝑗𝑁1𝑑1/2ξ€œ{𝑦>0,|π‘¦βˆ’π‘₯|<2π‘—βˆšπ‘‘}||||𝑓(𝑦)𝑑𝑦.(3.10)
Notice that, for 𝑗β‰₯0 and 𝑑>𝜌2𝐿𝛼(π‘₯π‘˜), we have 2π‘—βˆšπ‘‘β‰₯πœŒπΏπ›Ό(π‘₯)βˆΌπœŒπΏπ›Ό(π‘₯π‘˜), for π‘₯βˆˆπ΅βˆ—π‘˜. Thus 1βˆšπ‘‘ξ€œ{𝑦>0,|π‘¦βˆ’π‘₯|<2π‘—βˆšπ‘‘}||||𝑓(𝑦)𝑑𝑦≀𝐢2𝑗‖𝑓‖BMO𝐿𝛼.(3.11)
Therefore, sup𝑑>𝜌2𝐿𝛼π‘₯π‘˜ξ€Έ||𝑇𝛼𝑑||≲𝑓(π‘₯)βˆžξ“π‘—=012π‘βˆ’1‖𝑓‖BMO𝐿𝛼≲‖𝑓‖BMO𝐿𝛼.(3.12)
By Lemma 2.5, it suffices to show that sup0<π‘‘β‰€πœŒ2πΏπ›Όπ‘˜)(π‘₯|𝑇𝛼𝑑𝑓(π‘₯)| satisfies (2). Write 𝑓=π‘“πœ’{π‘₯π‘˜/2≀𝑦≀2π‘₯π‘˜}+π‘“πœ’{𝑦<π‘₯π‘˜/2}+π‘“πœ’{𝑦>2π‘₯π‘˜}=𝑓1+𝑓2+𝑓3.(3.13)
By Proposition 2.7, it easily follows that β€–β€–β€–β€–β€–sup0<π‘‘β‰€πœŒ2𝐿𝛼π‘₯π‘˜ξ€Έ||𝑇𝛼𝑑𝑓2||β€–β€–β€–β€–β€–(π‘₯)𝐿∞(π΅βˆ—π‘˜)≀𝐢‖𝑓‖BMO𝐿𝛼,β€–β€–β€–β€–β€–sup0<π‘‘β‰€πœŒ2𝐿𝛼π‘₯π‘˜ξ€Έ||𝑇𝛼𝑑𝑓3||β€–β€–β€–β€–β€–(π‘₯)𝐿∞(π΅βˆ—π‘˜)≀𝐢‖𝑓‖BMO𝐿𝛼.(3.14)
Indeed, since π‘₯∼π‘₯π‘˜, for π‘₯βˆˆπ΅βˆ—π‘˜, by (a) of Proposition 2.7 and Remark 2.8, we have ||𝑇𝛼𝑑𝑓2||β‰²ξ€œ(π‘₯)π‘₯π‘˜0/2𝑦𝛼+1/2π‘₯π‘˜βˆ’π›Όβˆ’3/2||||≲1𝑓(𝑦)𝑑𝑦π‘₯π‘˜ξ€œπ‘₯π‘˜0/2||𝑓||(𝑦)𝑑𝑦≲‖𝑓‖BMO𝐿𝛼.(3.15)
Similarly, ||𝑇𝛼𝑑𝑓3(||≲π‘₯)βˆžξ“π‘›=1ξ€·2𝑛π‘₯π‘˜ξ€Έβˆ’π›Όβˆ’3/2ξ€·π‘₯π‘˜ξ€Έπ›Ό+1/2ξ€œ2𝑛+1π‘₯π‘˜2𝑛π‘₯π‘˜||||≲𝑓(𝑦)π‘‘π‘¦βˆžξ“π‘›=1(2𝑛)βˆ’π›Όβˆ’1/2‖𝑓‖BMO𝐿𝛼≲‖𝑓‖BMO𝐿𝛼.(3.16)
Now, we come to treat 𝑓1. We make further decompositions. Split 𝑇𝛼𝑑𝑓1=𝑇𝛼𝑑𝑓1βˆ’π‘Šπ‘‘π‘“1ξ€Έ+ξ€·π‘Šπ‘‘π‘“1βˆ’π»π‘‘π‘“1ξ€Έ+𝐻𝑑𝑓1,(3.17)
where π»π‘‘ξ€œπ‘”(π‘₯)=∞0ξƒ¬π‘’βˆ’π‘‘πœ‹ξ€·1βˆ’π‘’βˆ’2𝑑1/2ξ‚΅βˆ’1exp21+π‘’βˆ’2𝑑1βˆ’π‘’βˆ’2𝑑(π‘₯βˆ’π‘¦)2𝑔(𝑦)𝑑𝑦.(3.18)
For the first term, by (c) of Proposition 2.7, we have ||𝑇𝛼𝑑𝑓1βˆ’π‘Šπ‘‘π‘“1||1≀𝐢π‘₯π‘˜ξ€œ2π‘₯π‘˜π‘₯2/2||||𝑓(𝑦)𝑑𝑦≀𝐢‖𝑓‖BMO𝐿𝛼.(3.19)
By (2.8), π‘Šπ‘‘π‘“1(π‘₯)βˆ’π»π‘‘π‘“1=ξ€œ(π‘₯)∞0ξƒ¬π‘’βˆ’π‘‘πœ‹ξ€·1βˆ’π‘’βˆ’2𝑑1/2ξ‚΅βˆ’1exp21+π‘’βˆ’2𝑑1βˆ’π‘’βˆ’2𝑑(π‘₯βˆ’π‘¦)2ξ‚Άξ‚ƒπ‘’βˆ’2π‘₯𝑦(1βˆ’π‘’βˆ’π‘‘)2/1βˆ’π‘’βˆ’2π‘‘ξ‚„π‘“βˆ’11(𝑦)𝑑𝑦.(3.20)
Notice that |π‘’βˆ’2π‘₯𝑦(1βˆ’π‘’βˆ’π‘‘)2/(1βˆ’π‘’βˆ’2𝑑)βˆ’1|≀𝑐𝑑π‘₯2π‘˜, when π‘‘β‰€πœŒπΏπ›Ό(π‘₯π‘˜)2, π‘₯π‘˜/2≀𝑦≀2π‘₯π‘˜ and π‘₯βˆˆπ΅βˆ—π‘˜. Therefore, for π‘₯βˆˆπ΅βˆ—π‘˜ and π‘‘β‰€πœŒπΏπ›Ό(π‘₯π‘˜)2, we obtain ||π‘Šπ‘‘π‘“1(π‘₯)βˆ’π»π‘‘π‘“1||(π‘₯)≲𝑑π‘₯2π‘˜ξ€œβˆž01βˆšπ‘‘π‘’βˆ’π‘0|π‘₯βˆ’π‘¦|2/𝑑||𝑓1||(𝑦)𝑑𝑦≲𝑑π‘₯2π‘˜ξ“1≀2π‘—β‰€πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ/βˆšπ‘‘2βˆ’π‘—(π‘βˆ’1)12π‘—βˆšπ‘‘ξ€œ{𝑦>0,|π‘¦βˆ’π‘₯|<2π‘—βˆšπ‘‘}||||𝑓(𝑦)𝑑𝑦+𝑑π‘₯2π‘˜ξ“πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ/βˆšπ‘‘<2𝑗2βˆ’π‘—(π‘βˆ’1)12π‘—βˆšπ‘‘ξ€œ{𝑦>0,|π‘¦βˆ’π‘₯|<2π‘—βˆšπ‘‘}||||≲𝑓(𝑦)𝑑𝑦1≀2π‘—β‰€πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ/βˆšπ‘‘2βˆ’π‘—(π‘βˆ’1)𝑑π‘₯2π‘˜ξƒ©ξƒ©πœŒ1+log𝐿𝛼π‘₯π‘˜ξ€Έ2π‘—βˆšπ‘‘ξƒͺξƒͺ‖𝑓‖BMO𝐿𝛼+βˆžξ“π‘—=02βˆ’π‘—(π‘βˆ’1)‖𝑓‖BMOπΏπ›Όβ‰²βˆžξ“π‘—=02βˆ’π‘—(π‘βˆ’1)‖𝑓‖BMO𝐿𝛼≲‖𝑓‖BMO𝐿𝛼.(3.21)
Finally, by Lemma 2.5 again, we need to show that sup0<π‘‘β‰€πœŒ2πΏπ›Όπ‘˜)(π‘₯|𝐻𝑑𝑓1(π‘₯)| satisfies (2). Consider 𝐡=π΅π‘Ÿ(π‘₯0)βŠ‚π΅βˆ—π‘˜ and write 𝑓1=𝑓1βˆ’π‘“π΅βˆ—ξ€Έπœ’π΅βˆ—+𝑓1βˆ’π‘“π΅βˆ—ξ€Έπœ’(π΅βˆ—)π‘βˆ©(0,∞)+π‘“π΅βˆ—πœ’(0,∞)ξ€»=𝑓11+𝑓12.(3.22) By Corollary 2.3, we choose a constant 𝐢𝐡=sup0<π‘‘β‰€πœŒπΏπ›Ό(π‘₯π‘˜)2|𝐻𝑑𝑓12(π‘₯0)|, 1||𝐡||ξ€œπ΅|||||sup0<π‘‘β‰€πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ2||𝐻𝑑𝑓1||(π‘₯)βˆ’πΆπ΅|||||1𝑑π‘₯≀||𝐡||ξ€œπ΅sup0<π‘‘β‰€πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ2||𝐻𝑑𝑓1(π‘₯)βˆ’π»π‘‘π‘“12ξ€·π‘₯0ξ€Έ||≀1𝑑π‘₯||𝐡||ξ€œπ΅sup0<π‘‘β‰€πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ2||𝐻𝑑𝑓11||1(π‘₯)𝑑π‘₯+||𝐡||ξ€œπ΅sup0<π‘‘β‰€πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ2||𝐻𝑑𝑓12(π‘₯)βˆ’π»π‘‘π‘“12ξ€·π‘₯0ξ€Έ||𝑑π‘₯.(3.23) For the first integral, by Corollary 2.4 it easily follows that 1||𝐡||ξ€œπ΅sup0<π‘‘β‰€πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ2||𝐻𝑑𝑓11||ξ‚΅1(π‘₯)𝑑π‘₯≀||𝐡||ξ€œπ΅||𝑓11||2𝑑π‘₯1/2≀𝐢‖𝑓‖BMO𝐿𝛼.(3.24) For the second integral, ||𝐻𝑑𝑓1(π‘₯)βˆ’π»π‘‘π‘“12ξ€·π‘₯0ξ€Έ||≀||||ξ€œ(π΅βˆ—)π‘βˆ©(0,∞)𝐻𝑑(π‘₯,𝑦)βˆ’π»π‘‘ξ€·π‘₯0𝑓,𝑦1(𝑦)βˆ’π‘“π΅ξ€Έ||||+||||ξ€œπ‘‘π‘¦βˆž0𝐻𝑑(π‘₯,𝑦)βˆ’π»π‘‘ξ€·π‘₯0𝑓,𝑦𝐡||||𝑑𝑦=𝐼𝑑1(π‘₯)+𝐼𝑑2(π‘₯).(3.25) By the mean value theorem and the elementary inequality 12π‘›π‘Ÿξ€œ{𝑦>0,|π‘¦βˆ’π‘₯0|<2π‘›π‘Ÿ}||𝑓(𝑦)βˆ’π‘“π΅||𝑑𝑦≀𝑐𝑛‖𝑓‖BMO𝐿𝛼,(3.26) we have 𝐼𝑑1ξ€œ(π‘₯)≲(π΅βˆ—)π‘βˆ©(0,∞)1βˆšπ‘‘π‘’βˆ’|π‘¦βˆ’π‘₯0|2/10𝑑||π‘₯βˆ’π‘₯0||βˆšπ‘‘||𝑓1(𝑦)βˆ’π‘“π΅||β‰²ξ€œπ‘‘π‘¦(π΅βˆ—)π‘βˆ©(0,∞)||π‘₯βˆ’π‘₯0||||π‘¦βˆ’π‘₯0||2||𝑓1(𝑦)βˆ’π‘“π΅||π‘‘π‘¦β‰²π‘Ÿβˆžξ“π‘›=01(2π‘›π‘Ÿ)2ξ€œ{𝑦>0,|π‘¦βˆ’π‘₯0|<2π‘›π‘Ÿ}ξ€·||𝑓(𝑦)βˆ’π‘“π΅||+||𝑓(𝑦)βˆ’π‘“1||≲(𝑦)π‘‘π‘¦βˆžξ“π‘›=02βˆ’π‘›(𝑛+1)‖𝑓‖BMO𝐿𝛼.(3.27) On the other hand, by the fact |𝑓𝐡|≀𝐢(1+log(πœŒπΏπ›Ό(π‘₯0)/π‘Ÿ))‖𝑓‖BMO𝐿𝛼 in Lemma 2.6, we obtain 𝐼𝑑2||||ξ€œ(π‘₯)≲0βˆ’βˆžξ€·π»π‘‘(π‘₯,𝑦)βˆ’π»π‘‘ξ€·π‘₯0𝑓,𝑦𝐡||||β‰²ξ€œπ‘‘π‘¦{𝑦<0}||π‘₯βˆ’π‘₯0||||π‘¦βˆ’π‘₯0||2||𝑓𝐡||β‰²π‘Ÿπ‘‘π‘¦πœŒπΏπ›Όξ€·π‘₯0ξ€Έξƒ©πœŒ1+log𝐿𝛼π‘₯0ξ€Έπ‘Ÿξƒͺ‖𝑓‖BMO𝐿𝛼≲‖𝑓‖BMO𝐿𝛼.(3.28) Therefore, we obtain 1||𝐡||ξ€œπ΅sup0<π‘‘β‰€πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ2||𝐻𝑑𝑓12(π‘₯)βˆ’π»π‘‘π‘“12ξ€·π‘₯0ξ€Έ||𝑑π‘₯≲‖𝑓‖BMO𝐿𝛼,(3.29) which establishes the proof.

4. 𝑔-Function

For all π‘“βˆˆπΏ1loc(0,∞) and π‘₯∈(0,∞), define the Littlewood-Paley 𝑔-function by ξ‚΅ξ€œπ‘”(𝑓)(π‘₯)β‰‘βˆž0||𝑄𝑑||𝑓(π‘₯)2𝑑𝑑𝑑1/2,(4.1) where, {𝑄𝑑}𝑑>0 is a family of operators with the integral kernels 𝑄𝑑(π‘₯,𝑦)=𝑑2πœ•π‘Šπ›Όπ‘ (π‘₯,𝑦)||||πœ•π‘ π‘ =𝑑2.(4.2)

Theorem 4.1. Let 𝛼>βˆ’1/2. There exists a constant 𝐢>0 such that, for all π‘“βˆˆBMO𝐿𝛼, 𝑔(𝑓)∈BMO𝐿𝛼 and ‖𝑔(𝑓)β€–BMO𝐿𝛼≀𝐢‖𝑓‖BMO𝐿𝛼.

Proof. By Proposition 2.9 and (a) of Proposition 2.10, we have 𝑄𝑑1(π‘₯,𝑦)β‰€π‘π‘‘π‘’βˆ’π‘1|π‘₯βˆ’π‘¦|2/𝑑2.(4.3) For π‘“βˆˆBMO𝐿𝛼, because of this and the integrability of (1+|π‘₯|)βˆ’2𝑓(π‘₯) (see [12, page 141]), π‘„π‘‘ξ€œπ‘“(π‘₯)=∞0𝑄𝑑(π‘₯,𝑦)𝑓(𝑦)𝑑𝑦(4.4) is well defined absolutely convergent integral for all (π‘₯,𝑑)∈(0,∞)Γ—(0,∞). Similar to the proof of Theorem 3.2, we will try to show that, for π΅π‘˜βŠ‚β„¬ in Lemma 2.1, (1)(1/|π΅π‘˜βˆ«|)π΅π‘˜|𝑔(𝑓)(π‘₯)|𝑑π‘₯≀𝐢‖𝑓‖BMO𝐿𝛼, (2)‖𝑔(𝑓)(π‘₯)β€–BMO(π΅βˆ—π‘˜)≀𝐢‖𝑓‖BMO𝐿𝛼. We split []𝑔(𝑓)(π‘₯)2=𝑔1ξ€»(𝑓)(π‘₯)2+𝑔2ξ€»(𝑓)(π‘₯)2=ξ€œ20πœŒπΏπ›Ό(π‘₯π‘˜)0||𝑄𝑑||𝑓(π‘₯)2𝑑𝑑𝑑+ξ€œβˆž20πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ||𝑄𝑑||𝑓(π‘₯)2𝑑𝑑𝑑.(4.5) By Lemma 2.11 and HΓΆlder inequality, assertion (1) holds for 𝑔1(𝑓)(π‘₯). To finish the proof of (1), it suffices to show that ‖‖𝑔2β€–β€–(𝑓)𝐿∞(π΅βˆ—π‘˜)≀𝑐‖𝑓‖BMO𝐿𝛼.(4.6)
In the next proof, for the sake of brevity we introduce the additional notations: 𝑋𝑑1𝑒(π‘₯)=π‘¦βˆˆ(0,∞)βˆΆβˆ’π‘‘2π‘₯𝑦1βˆ’π‘’βˆ’2𝑑2𝑋≀1,(4.7)𝑑2𝑒(π‘₯)=&π‘¦βˆˆ(0,∞)βˆΆβˆ’π‘‘2π‘₯𝑦1βˆ’π‘’βˆ’2𝑑2ξƒ°>1.(4.8) By 𝑋𝑑1(π‘₯) and 𝑋𝑑1(π‘₯), we split 𝑄𝑑𝑓(π‘₯) as ||𝑄𝑑||β‰€ξ€œπ‘“(π‘₯)𝑋𝑑1(π‘₯)||||||𝑄𝑓(𝑦)𝑑(||ξ€œπ‘₯,𝑦)𝑑𝑦+𝑋𝑑2(π‘₯)||||||𝑄𝑓(𝑦)𝑑(π‘₯,𝑦)βˆ’π‘ƒπ‘‘(||+ξ€œπ‘₯,𝑦)𝑑𝑦𝑋𝑑2(π‘₯)||𝑓||||𝑃(𝑦)𝑑||(π‘₯,𝑦)𝑑𝑦=𝐼𝑑1(π‘₯)+𝐼𝑑2(π‘₯)+𝐼𝑑3(π‘₯).(4.9)
For 𝐼𝑑1(π‘₯) and π‘₯βˆˆπ΅βˆ—π‘˜, we shall first show the inequality 𝐽1ξ€œ(π‘₯)=∞20πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ||𝐼𝑑1||𝑓(π‘₯)2𝑑𝑑𝑑≀𝑐‖𝑓‖2BMO𝐿𝛼.(4.10) Using (a) of Proposition 2.9, if π‘₯π‘˜β‰€1, πœŒπΏπ›Ό(π‘₯π‘˜)∼π‘₯π‘˜, we get ||𝑄𝑑(||π‘₯,𝑦)≀𝐢𝑑2𝛼+3ξ‚€π‘₯𝑦𝑑2𝛼+1/2π‘’βˆ’(π‘₯2+𝑦2)/8𝑑2π‘’βˆ’(𝛼+1)𝑑2ξ€·1βˆ’π‘’βˆ’2𝑑2𝛼+2ξ‚€π‘₯β‰€πΆπ‘˜π‘‘ξ‚π›Ό+1/21π‘‘π‘’βˆ’|π‘₯βˆ’π‘¦|2/16𝑑2.(4.11)
If π‘₯π‘˜β‰₯1, πœŒπΏπ›Ό(π‘₯π‘˜)∼1/π‘₯π‘˜, we have ||𝑄𝑑||1(π‘₯,𝑦)β‰€πΆπ‘‘π‘’βˆ’(π‘₯2+𝑦2)/8𝑑2π‘’βˆ’((𝛼+1)/2)𝑑21≀𝐢𝑑π‘₯π‘˜1π‘‘π‘’βˆ’|π‘₯βˆ’π‘¦|2/16𝑑2.(4.12) The previous two inequalities above imply 𝐽1ξ€œ(π‘₯)β‰²βˆž20πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ|||||ξ€œβˆž0ξƒ©πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έπ‘‘ξƒͺ𝜎1π‘‘π‘’βˆ’|π‘₯βˆ’π‘¦|2/16𝑑2|||||||||𝑓(𝑦)𝑑𝑦2π‘‘π‘‘π‘‘β‰²ξ€œβˆž20πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έξƒ©πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έπ‘‘ξƒͺ2𝜎|||||βˆžξ“π‘—=02βˆ’π‘—(π‘βˆ’1)12π‘—π‘‘ξ€œ{𝑦>0,|π‘¦βˆ’π‘₯|<2𝑗𝑑}|||||||||𝑓(𝑦)𝑑𝑦2𝑑𝑑𝑑≲‖𝑓‖2BMO𝐿𝛼.(4.13)
For 𝐼𝑑2(π‘₯) and π‘₯βˆˆπ΅βˆ—π‘˜, we shall also prove the inequality 𝐽2ξ€œ(π‘₯)=∞20πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ||𝐼𝑑2||𝑓(π‘₯)2𝑑𝑑𝑑≀𝑐‖𝑓‖2BMO𝐿𝛼.(4.14) We split this integral as 𝐽2ξ€œ(π‘₯)=120πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ||𝐼𝑑2||𝑓(π‘₯)2𝑑𝑑𝑑+ξ€œβˆž1||𝐼𝑑2||𝑓(π‘₯)2𝑑𝑑𝑑=𝐽3(π‘₯)+𝐽4(π‘₯).(4.15) To deal with 𝐽3(π‘₯), we discuss two cases. In the first case of π‘₯π‘˜β‰€1, notice that 𝑦>π‘₯, when π‘₯βˆˆπ΅βˆ—π‘˜, π‘¦βˆˆπ‘‹π‘‘2(π‘₯π‘˜) and 𝑑β‰₯20πœŒπΏπ›Ό(π‘₯π‘˜). According to (b) of Proposition 2.9, 𝐽3ξ€œ(π‘₯)≲120πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ|||||ξ€œβˆžξ‚€1βˆ’π‘’2βˆ’2𝑑𝑒𝑑2/π‘₯1π‘‘π‘’βˆ’|π‘₯βˆ’π‘¦|2/4𝑑2π‘₯𝑑1βˆ’π‘’βˆ’2𝑑2𝑒𝑑2βˆ’π‘₯2|||||||||𝑓(𝑦)𝑑𝑦2π‘‘π‘‘π‘‘β‰²ξ€œβˆž20πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έπ‘₯2π‘˜π‘‘2||||ξ€œβˆž01π‘‘π‘’βˆ’|π‘₯βˆ’π‘¦|2/4𝑑2||||||||𝑓(𝑦)𝑑𝑦2𝑑𝑑𝑑≲‖𝑓‖2BMO𝐿𝛼.(4.16) The last inequality is from the same proof of 𝐽1(π‘₯). In the second case of π‘₯π‘˜>1, using (b) of Proposition 2.9 again, for 𝑑>20πœŒπΏπ›Ό(π‘₯π‘˜) we obtain ||𝐼𝑑2||≲1(π‘₯)π‘₯π‘˜ξ€œ2π‘₯π‘˜π‘₯π‘˜/2𝑑π‘₯π‘˜||||ξ€œπ‘“(𝑦)𝑑𝑦+(0,∞)∩(π‘₯π‘˜/2,2π‘₯π‘˜)𝑐1π‘‘π‘’βˆ’|π‘₯βˆ’π‘¦|2/2𝑑2||||≲𝑑𝑓(𝑦)𝑑𝑦π‘₯π‘˜β€–π‘“β€–BMO𝐿𝛼+𝑑π‘₯π‘˜ξ€œβˆž01π‘‘π‘’βˆ’|π‘₯βˆ’π‘¦|2/4𝑑2||||≲𝑑𝑓(𝑦)𝑑𝑦π‘₯π‘˜β€–π‘“β€–BMO𝐿𝛼.(4.17) The last inequality is also from the same proof of 𝐽1(π‘₯). Inserting this into 𝐽3(π‘₯) leads to 𝐽3(π‘₯)≀𝐢‖𝑓‖2BMO𝐿𝛼. Now, it remains to show 𝐽4(π‘₯)≀𝑐‖𝑓‖2BMO𝐿𝛼. Using (b) of Proposition 2.9, by the standard argument it easily follows that 𝐽4ξ€œ(π‘₯)β‰²βˆž1π‘’βˆ’π‘‘/10𝑑||||ξ€œβˆž01π‘‘π‘’βˆ’|π‘₯βˆ’π‘¦|2/2𝑑2||||||||𝑓(𝑦)𝑑𝑦2𝑑𝑑≲‖𝑓‖2BMO𝐿𝛼.(4.18) To complete the proof of (4.6), we need to show that 𝐽5ξ€œ(π‘₯)=∞20πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ||𝐼𝑑3||𝑓(π‘₯)2𝑑𝑑𝑑≀𝑐‖𝑓‖2BMO𝐿𝛼.(4.19) We also consider two cases of π‘₯π‘˜β‰€1 and π‘₯π‘˜>1. When π‘₯π‘˜β‰€1, repeating the above argument for 𝐽3(π‘₯) and using (a) of Proposition 2.10, we have 𝐽5≀𝑐‖𝑓‖2BMO𝐿𝛼. When π‘₯π‘˜>1, using (a) of Proposition 2.10 again, for 𝑑β‰₯20πœŒπΏπ›Ό(π‘₯π‘˜), we obtain ||𝐼𝑑3||≲1(π‘₯)π‘₯π‘˜π‘‘ξ€œβˆž01π‘‘π‘’βˆ’π‘|π‘¦βˆ’π‘₯|2/𝑑2||||≲1𝑓(𝑦)𝑑𝑦𝑑π‘₯π‘˜βˆžξ“π‘—=02βˆ’π‘—(π‘βˆ’1)12π‘—π‘‘ξ€œ{𝑦>0,|π‘¦βˆ’π‘₯|<2𝑗𝑑}||||β‰²πœŒπ‘“(𝑦)𝑑𝑦𝐿𝛼π‘₯π‘˜ξ€Έπ‘‘β€–π‘“β€–BMO𝐿𝛼,(4.20) which shows that (4.19) holds.
Next, we come to prove assertion (2). By (4.6) and Lemma 2.5, we only need to show β€–β€–β€–β€–ξ‚΅ξ€œ20πœŒπΏπ›Ό(π‘₯π‘˜)0||𝑄𝑑||𝑓(π‘₯)2𝑑𝑑𝑑1/2β€–β€–β€–β€–BMO(π΅βˆ—π‘˜)≀𝐢‖𝑓‖BMO𝐿𝛼.(4.21)
Consider any ball 𝐡=π΅π‘Ÿ(π‘₯0)βŠ‚π΅βˆ—π‘˜. By Lemma 2.11, we have 1||𝐡||ξ€œπ΅ξ‚΅ξ€œπ‘Ÿ0||𝑄𝑑||𝑓(π‘₯)2𝑑𝑑𝑑1/2ξ‚΅1𝑑π‘₯≀||𝐡||ξ€œπ΅ξ€œπ‘Ÿ0||𝑄𝑑||𝑓(π‘₯)2𝑑𝑑𝑑𝑑π‘₯1/2≀𝐢‖𝑓‖BMO𝐿𝛼.(4.22) Therefore, by Lemma 2.5 and Corollary 2.3, it suffices to prove 1||𝐡||ξ€œπ΅|||||ξ‚΅ξ€œ20πœŒπΏπ›Ό(π‘₯π‘˜)π‘Ÿ||𝑄𝑑||𝑓(π‘₯)2𝑑𝑑𝑑1/2βˆ’ξ‚΅ξ€œ20πœŒπΏπ›Ό(π‘₯π‘˜)π‘Ÿ||𝑄𝑑𝑓π‘₯0ξ€Έ||2𝑑𝑑𝑑1/2|||||𝑑π‘₯≀𝐢‖𝑓‖BMO𝐿𝛼.(4.23) To prove (4.23), we first claim that, for all π‘“βˆˆBMO𝐿𝛼, π‘₯βˆˆπ΅βˆ—π‘˜, and 𝑑≀20πœŒπΏπ›Ό(π‘₯π‘˜), ||𝑄𝑑||𝑓(π‘₯)≀𝐢‖𝑓‖BMO𝐿𝛼.(4.24)
We shall split into three different estimates: ξ€œπ‘‹π‘‘1(π‘₯)||||||𝑄𝑓(𝑦)𝑑(||π‘₯,𝑦)𝑑𝑦≀𝐢‖𝑓‖BMO𝐿𝛼,ξ€œ(4.25)𝑋𝑑2(π‘₯)||𝑓||||𝑄(𝑦)𝑑(π‘₯,𝑦)βˆ’π‘ƒπ‘‘||(π‘₯,𝑦)𝑑𝑦≀𝐢‖𝑓‖BMO𝐿𝛼,||||ξ€œ(4.26)𝑋𝑑2(π‘₯)𝑓(𝑦)𝑃𝑑||||(π‘₯,𝑦)𝑑𝑦≀𝐢‖𝑓‖BMO𝐿𝛼.(4.27) Let us first treat (4.25). Since 𝑦≀𝑐(πœŒπΏπ›Ό(π‘₯π‘˜)2/π‘₯π‘˜), when π‘¦βˆˆπ‘‹π‘‘1(π‘₯), notice that π‘₯∼π‘₯π‘˜ when π‘₯βˆˆπ΅βˆ—π‘˜, using (a) of Proposition 2.9, and recalling the definition of πœŒπΏπ›Ό(π‘₯), we have ξ€œπ‘‹π‘‘1(π‘₯)||||||𝑄𝑓(𝑦)𝑑||ξ€œ(π‘₯,𝑦)π‘‘π‘¦β‰€πΆπ‘πœŒπΏπ›Ό(π‘₯π‘˜)2/π‘₯π‘˜0𝑑π‘₯2π‘˜||||𝑑𝑓(𝑦)𝑑𝑦≀𝐢π‘₯π‘˜β€–π‘“β€–BMO𝐿𝛼≀𝐢‖𝑓‖BMO𝐿𝛼.(4.28) For (4.26), using (b) of Proposition 2.9, for π‘₯βˆˆπ΅βˆ—π‘˜, the left side of (4.26) is controlled by 1π‘₯π‘˜ξ€œ2π‘₯π‘˜π‘₯π‘˜/2𝑑π‘₯π‘˜||||ξ€œπ‘“(𝑦)𝑑𝑦+(0,∞)∩(π‘₯π‘˜/2,2π‘₯π‘˜)𝑐1π‘‘π‘’βˆ’|π‘₯βˆ’π‘¦|2/2𝑑2||||≲𝑑𝑓(𝑦)𝑑𝑦π‘₯π‘˜β€–π‘“β€–BMO𝐿𝛼+𝑑π‘₯π‘˜ξ€œβˆž01π‘‘π‘’βˆ’|π‘₯βˆ’π‘¦|2/4𝑑2||||≲𝑑𝑓(𝑦)𝑑𝑦π‘₯π‘˜β€–π‘“β€–BMO𝐿𝛼+𝑑π‘₯π‘˜ξ‚Ά12‖𝑓‖BMO𝐿𝛼.(4.29) The third inequality is from the same argument for dealing with (3.21) in the proof of Theorem 3.2.
For (4.27), we write ||||ξ€œπ‘‹π‘‘2(π‘₯)𝑓(𝑦)𝑃𝑑||||≀||||ξ€œ(π‘₯,𝑦)𝑑𝑦𝑋𝑑2(π‘₯)||𝑓(𝑦)βˆ’π‘“π΅π‘‘(π‘₯)||||𝑃𝑑||||||+||||ξ€œ(π‘₯,𝑦)𝑑𝑦𝑋𝑑2(π‘₯)𝑓𝐡𝑑(π‘₯)𝑃𝑑||||.(π‘₯,𝑦)𝑑𝑦(4.30)
By (a), (c) of Proposition 2.10 and the fact that |𝑓𝐡𝑑(π‘₯)|≲(1+log20πœŒπΏπ›Ό(π‘₯π‘˜)/𝑑)‖𝑓‖BMO𝐿𝛼, we have ||||ξ€œπ‘‹π‘‘2(π‘₯)𝑓𝐡𝑑(π‘₯)𝑃𝑑||||≀||||ξ€œ(π‘₯,𝑦)π‘‘π‘¦βˆžβˆ’βˆžπ‘“π΅π‘‘(π‘₯)𝑃𝑑||||+|||||ξ€œ(π‘₯,𝑦)𝑑𝑦𝑒𝑑2(1βˆ’π‘’2βˆ’2𝑑)/π‘₯βˆ’βˆžπ‘“π΅π‘‘(π‘₯)𝑃𝑑|||||≲𝑑(π‘₯,𝑦)𝑑𝑦2πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ21+log20πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έπ‘‘ξƒͺ‖𝑓‖BMO𝐿𝛼+ξ€œ(1βˆ’π‘’2βˆ’2𝑑)𝑒𝑑2/π‘₯βˆ’βˆž1π‘‘π‘’βˆ’(π‘₯βˆ’π‘¦)2/16𝑑2𝑑𝑦1+log20πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έπ‘‘ξƒͺ‖𝑓‖BMOπΏπ›Όβ‰²ξ€œ(1βˆ’π‘’2βˆ’2𝑑)𝑒𝑑2/𝑑π‘₯βˆ’π‘₯/π‘‘βˆ’βˆžπ‘’(βˆ’1/16)𝑧2𝑑𝑧1+log20πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έπ‘‘ξƒͺ‖𝑓‖BMO𝐿𝛼+‖𝑓‖BMOπΏπ›Όβ‰²π‘’βˆ’(1/20)((1βˆ’π‘’2βˆ’2𝑑)𝑒𝑑2/𝑑π‘₯βˆ’π‘₯𝑑)1+log20πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έπ‘‘ξƒͺ‖𝑓‖BMO𝐿𝛼+‖𝑓‖BMO𝐿𝛼≲𝑑2πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ21+log20πœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έπ‘‘ξƒͺ‖𝑓‖BMO𝐿𝛼+‖𝑓‖BMO𝐿𝛼≲‖𝑓‖BMO𝐿𝛼.(4.31) The third and fourth inequalities come from the fact (1βˆ’π‘’βˆ’2𝑑2)𝑒𝑑2/π‘₯𝑑≲π‘₯/𝑑 and π‘₯∼π‘₯π‘˜, when 𝑑≀20πœŒπΏπ›Ό(π‘₯π‘˜) and π‘₯βˆˆπ΅βˆ—π‘˜. It is notable that, for 𝑑≀20πœŒπΏπ›Ό(π‘₯π‘˜) and π‘₯βˆˆπ΅βˆ—π‘˜, the inequality above also implies ||||ξ€œπ‘‹π‘‘2(π‘₯)𝑃𝑑||||𝑑(π‘₯,𝑦)π‘‘π‘¦β‰€πΆπœŒπΏπ›Όξ€·π‘₯π‘˜ξ€Έ.(4.32) On the other hand, by (a) of Proposition 2.10 and the simple fact that |𝑓2𝑗𝐡𝑑(π‘₯)βˆ’π‘“π΅π‘‘(π‘₯)|≀𝐢𝑗‖𝑓‖BMO𝐿𝛼, we obtain ||||ξ€œπ‘‹π‘‘2(π‘₯)||𝑓(𝑦)βˆ’π‘“π΅π‘‘(π‘₯)||||𝑃𝑑||||||β‰²ξ€œ(π‘₯,𝑦)π‘‘π‘¦βˆž01π‘‘π‘’βˆ’|π‘¦βˆ’π‘₯|2/10𝑑2||𝑓(𝑦)βˆ’π‘“π΅π‘‘(π‘₯)||β‰²π‘‘π‘¦βˆžξ“π‘—=02βˆ’π‘—(π‘βˆ’1)12π‘—π‘‘ξ€œ{𝑦>0,|π‘¦βˆ’π‘₯|≀2𝑗𝑑}||𝑓(𝑦)βˆ’π‘“2𝑗𝐡𝑑(π‘₯)||+π‘‘π‘¦βˆžξ“π‘—=02βˆ’π‘—(π‘βˆ’1)||𝑓2𝑗𝐡𝑑(π‘₯)βˆ’π‘“π΅π‘‘(π‘₯)||β‰²βˆžξ“π‘—=0𝑗2βˆ’π‘—(π‘βˆ’1)‖𝑓‖BMO𝐿𝛼≲‖𝑓‖BMO𝐿𝛼.(4.33) From (4.24), we deduce that |||||ξ‚΅ξ€œ20πœŒπΏπ›Ό(π‘₯π‘˜)π‘Ÿ||𝑄𝑑||𝑓(π‘₯)2𝑑𝑑𝑑1/2βˆ’ξ‚΅ξ€œ20πœŒπΏπ›Ό(π‘₯π‘˜)π‘Ÿ||𝑄𝑑𝑓π‘₯0ξ€Έ||2𝑑𝑑𝑑1/2|||||β‰²ξ‚΅ξ€œ20πœŒπΏπ›Ό(π‘₯π‘˜)π‘Ÿ||𝑄𝑑𝑓(π‘₯)βˆ’π‘„π‘‘π‘“ξ€·π‘₯0ξ€Έ||2𝑑𝑑𝑑1/2≲‖𝑓‖1/2BMOπΏπ›Όξ‚΅ξ€œ20πœŒπΏπ›Ό(π‘₯π‘˜)π‘Ÿ||𝑄𝑑𝑓(π‘₯)βˆ’π‘„π‘‘π‘“ξ€·π‘₯0ξ€Έ||𝑑𝑑𝑑1/2.(4.34) By (4.28) and (4.29), for π‘§βˆˆπ΅βˆ—π‘˜, we have ξ€œ20πœŒπΏπ›Ό(π‘₯π‘˜)π‘Ÿξ€œπ‘‹π‘‘1(𝑧)||||||𝑄𝑓(𝑦)𝑑||(𝑧,𝑦)𝑑𝑦𝑑𝑑𝑑≀𝐢‖𝑓‖BMO𝐿𝛼,ξ€œ20πœŒπΏπ›Ό(π‘₯π‘˜)π‘Ÿξ€œπ‘‹π‘‘2(𝑧)||||||𝑄𝑓(𝑦)𝑑(𝑧,𝑦)βˆ’π‘ƒπ‘‘||(𝑧,𝑦)𝑑𝑦𝑑𝑑𝑑≀𝐢‖𝑓‖BMO𝐿𝛼.(4.35) To finish the proof of (4.23), it suffices to show ξ€œ20πœŒπΏπ›Ό(π‘₯π‘˜)π‘Ÿ||||ξ€œπ‘‹π‘‘2(π‘₯)𝑃𝑑(π‘₯,𝑦)βˆ’π‘ƒπ‘‘ξ€·π‘₯0𝑓||||,𝑦(𝑦)𝑑𝑦𝑑𝑑𝑑≀𝐢‖𝑓‖BMO𝐿𝛼,ξ€œ(4.36)20πœŒπΏπ›Ό(π‘₯π‘˜)π‘Ÿ||||ξ€œπ‘‹π‘‘2(π‘₯)𝑃𝑑π‘₯0ξ€Έπ‘“ξ€œ,𝑦(𝑦)π‘‘π‘¦βˆ’π‘‹π‘‘2(π‘₯0)𝑃𝑑π‘₯0𝑓||||,𝑦(𝑦)𝑑𝑦𝑑𝑑𝑑≀𝐢‖𝑓‖BMO𝐿𝛼.(4.37) It is easy to check (4.37). Indeed, without loss of generality, we assume π‘₯<π‘₯0. According to (a) of Proposition 2.10, ||||ξ€œπ‘‹π‘‘2(π‘₯)𝑃𝑑π‘₯0ξ€Έπ‘“ξ€œ,𝑦(𝑦)π‘‘π‘¦βˆ’π‘‹π‘‘2(π‘₯0)𝑃𝑑π‘₯0ξ€Έ||||β‰²ξ€œ,𝑦𝑓(𝑦)𝑑𝑦(1βˆ’π‘’2βˆ’2𝑑)