Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 923874 | 21 pages | https://doi.org/10.1155/2012/923874

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

Academic Editor: Aurelian Gheondea
Received29 Feb 2012
Revised21 May 2012
Accepted21 Jun 2012
Published13 Aug 2012

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