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

Abstract

Let {𝜑^𝛼𝑛}_𝑛∈ℕ be the Laguerre functions of Hermite type with index 𝛼. These are eigenfunctions of the Laguerre differential operator 𝐿_𝛼=1/2(−𝑑²/𝑑𝑦²+𝑦²+1/𝑦²(𝛼²−1/4)). 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 𝐿_𝛼 in the localized BMO space BMO_𝐿𝛼, which is the dual space of the Hardy space 𝐻^1𝐿_𝛼.

1. Introduction

Let 𝑛∈ℕ, 𝛼>−1. The Laguerre function of Hermite type 𝜑_𝛼 on (0,∞) is defined as 𝜑^𝛼𝑛(𝑦)=Γ(𝑛+1)Γ(𝑛+1+𝛼)^1/2𝑒^−𝑦2/2𝑦^𝛼𝐿^𝛼𝑛𝑦²(2𝑦)^1/2,𝑦∈(0,∞),(1.1) where 𝐿^𝛼𝑛(𝑥) denotes the Laguerre polynomial of degree 𝑛 and order 𝛼, see [1]. It is well known that for every 𝛼>−1 the system {𝜑^𝛼𝑛}^∞𝑛=0 forms an orthonormal basis of 𝐿²(0,∞). Moreover, these functions are eigenfunctions of the Laguerre differential operator 𝐿_𝛼=12−𝑑𝑑𝑦²²+𝑦²+1𝑦²𝛼²−14(1.2) satisfying 𝐿_𝛼𝜑^𝛼𝑛=(2𝑛+𝛼+1)𝜑^𝛼𝑛. The operator 𝐿_𝛼 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 𝐿_𝛼. Let {𝑇^𝛼𝑡}_𝑡≥0 be the heat-diffusion semigroup generated by −𝐿_𝛼. More precisely, for 𝑓∈𝐿²(0,∞), we define 𝑇^𝛼𝑡𝑓(𝑥)=^∞0𝑊^𝛼𝑡(𝑥,𝑦)𝑓(𝑦)𝑑𝑦,(1.3) where 𝑊^𝛼𝑡(𝑥,𝑦)=2𝑒^−𝑡1−𝑒^−2𝑡^1/212𝑥−𝑦𝑒𝑒^{−𝑡−2𝑡}^1/2𝐼_𝛼12𝑥−𝑦𝑒𝑒^{−𝑡−2𝑡}exp−1211+−𝑒𝑒^{−2𝑡−2𝑡}𝑥²+𝑦².(1.4)𝐼_𝛼 is the modified Bessel function of the first kind and order 𝛼.

In [3], we introduced and developed a localized BMO space BMO_𝐿𝛼 associated with the operator 𝐿_𝛼, which is the dual space of the Hardy space 𝐻^1𝐿_𝛼 introduced by Dziubański [4]. More precisely, let 𝜌_𝐿𝛼(𝑥)=18min𝑥,𝑥1,𝑥>0.(1.5)

Definition 1.1. Let 𝛼>−1/2, 𝐵_𝑠(𝑦) be any ball in (0,∞) with the center 𝑦 and the radius 𝑠 and 𝑓 a locally integrable function on (0,∞). We say 𝑓∈BMO_𝐿𝛼 if there exists a constant 𝐶≥0 independent of 𝑠 and 𝑦 such that 1||𝐵_𝑠(𝑦)||_{𝐵𝑠(𝑦)}1||𝑓−𝑓_{𝐵𝑠(𝑦)}||≤𝐶,if𝑠<𝜌_𝐿𝛼(𝑦),||𝐵_𝑠(𝑦)||_{𝐵𝑠(𝑦)}||𝑓||≤𝐶,if𝑠≥𝜌_𝐿𝛼(𝑦).(1.6) Here, 𝑓_{𝐵𝑠(𝑦)}=(1/|𝐵_𝑠(𝑦)|)∫_{𝐵𝑠(𝑦)}𝑓𝑑𝑥. We let ‖𝑓‖_BMO𝐿𝛼 denote the smallest 𝐶 in the two inequalities above.

It is readily seen that BMO_𝐿𝛼 is a Banach space with norm ‖⋅‖_BMO𝐿𝛼.

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

These results were investigated by Dziubański et al. in [5] for Schrödinger operators on ℝ^𝑑 with 𝑑≥3 and with potentials satisfying a reverse Hölder'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(𝑥)≤𝐶⁰1𝜌(𝑦)1+𝑑𝜌(𝑥(𝑦,)𝑦)^𝑘0.(1.7) Obviously, our 𝜌_𝐿𝛼 in (1.5) does not satisfy this condition. Indeed, let 𝑥 tend to zero and 𝑦=1; then the left side becomes greater than the right.

It is notable the generalized square functions associated to Schrödinger 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 BMO_𝐿𝛼 to BMO_𝐿𝛼. In Section 4, we obtain the boundedness on BMO_𝐿𝛼 of the Littlewood-Paley 𝑔-function associated with the heat semigroup for 𝐿_𝛼. We make some conventions. Throughout this paper by 𝐶 we always denote a positive constant that may vary at each occurrence; 𝐵_𝑟(𝑦₀) stands for {𝑦>0,|𝑦−𝑦₀|≤𝑟}; 𝐴∼𝐵 means (1/𝐶)𝐴≤𝐵≤𝐶𝐴, and the notation 𝑋≲𝑌 is used to indicate that 𝑋≤𝐶𝑌 with an independent positive constant 𝐶.

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 𝑥₀=1,   𝑥_𝑗=𝑥_𝑗−1+𝜌_𝐿𝛼(𝑥_𝑗−1) for 𝑗≥1, and 𝑥_𝑗=𝑥_𝑗+1−𝜌_𝐿𝛼(𝑥_𝑗+1) for 𝑗<0. One defines the family of “critical balls” of ℬ={𝐵_𝑘}^{∞𝑘=−∞}, where 𝐵_𝑘∶={𝑥∈(0,∞)∶|𝑥−𝑥_𝑘|<𝜌_𝐿𝛼(𝑥_𝑘)}. Then one has(a)⋃^{∞𝑘=−∞}𝐵_𝑘=(0,∞), (b)for every 𝑘∈ℤ, 𝐵_𝑘∩𝐵_𝑗=∅ provided that 𝑗∉{𝑘−1,𝑘,𝑘+1}, (c)for any 𝑦₀∈(0,∞), at most three balls in ℬ have nonempty intersection with 𝐵(𝑦₀,𝜌_𝐿𝛼(𝑦₀)).

Corollary 2.2. There exists a constant 𝐶>0 such that for every 𝐵_𝑅(𝑥)⊆(0,∞) with 𝑅>𝜌_𝐿𝛼(𝑥), one has ||𝐵_𝑅(𝑥)||≤_{𝐵𝑘∈ℬ∶𝐵𝑘∩𝐵𝑅(𝑥)≠∅}||𝐵_𝑘||≤𝐶||𝐵_𝑅(𝑥)||.(2.1)

Corollary 2.3. There exists a constant 𝐶 such that, for 𝑓∈BMO_𝐿𝛼, one has ‖𝑓‖_BMO𝐿𝛼≤𝐶sup_𝑘||𝑓||^𝐵_𝑘+‖𝑓‖_BMO(𝐵^∗𝑘₎,(2.2) where, for any ball 𝐵, the norm ‖⋅‖_BMO(𝐵) is given by ‖𝑓‖_BMO(𝐵)=sup_{𝐵𝑟(𝑥)⊂𝐵}1𝐵_𝑟(𝑥)_{𝐵𝑟(𝑥)}∼sup_{𝐵𝑟(𝑥)⊂𝐵}||𝑓−𝑓_{𝐵𝑟(𝑥)}||𝑑𝑦inf_𝑐∈ℂ1𝐵_𝑟(𝑥)_{𝐵𝑟(𝑥)}||𝑓−𝑐||𝑑𝑦.(2.3)

Corollary 2.4 (see [3, Corollary  3]). Let 𝐵=𝐵_𝑟(𝑦₀)⊂(0,∞). There exists a constant 𝐶>0 such that, for all 𝑓∈BMO_𝐿𝛼, one has (1)if 𝑟≥𝜌_𝐿𝛼(𝑦₀)/2, then ((1/|𝐵|)∫_𝐵|𝑓(𝑥)|²𝑑𝑥)^1/2≤𝐶‖𝑓‖_BMO𝐿𝛼, (2)if 𝑟<𝜌_𝐿𝛼(𝑦₀)/2, then ((1/|𝐵|)∫_𝐵|𝑓(𝑥)−𝑓_𝐵|²𝑑𝑥)^1/2≤𝐶‖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 ℎ∈BMO(𝐵^∗𝑘) and 𝑔₁ and 𝑔₂ be functions in 𝐿^∞(0,∞). If 𝑓 is any measurable function satisfying ℎ+𝑔₁≤𝑓≤ℎ+𝑔₂,a.e.,(2.4) then 𝑓∈BMO(𝐵^∗𝑘) and ‖𝑓‖_BMO(𝐵^∗𝑘₎≤‖ℎ‖_BMO(𝐵^∗𝑘₎+max(‖𝑔₁‖_∞,‖𝑔₂‖_∞).

Lemma 2.6. For all 𝑓∈BMO_𝐿𝛼 and 𝐵=𝐵_𝑟(𝑦₀) with 𝑟<𝜌_𝐿𝛼(𝑦₀). There exists a constant 𝐶>0 such that ||𝑓_𝐵∗||≤𝐶1+log𝜌^𝐿𝛼𝑦₀𝑟‖𝑓‖^BMO𝐿𝛼.(2.5)

Let 𝐻 be the Hermite operator 𝐻=12−𝑑𝑑𝑥²²+𝑥².(2.6) One considers the heat diffusion semigroup {𝑊_𝑡}_𝑡>0 associated with 𝐻 and defined by, for every 𝑓∈𝐿²(ℝ), 𝑊_𝑡𝑓(𝑥)=_ℝ𝑊_𝑡(𝑥,𝑦)𝑓(𝑦)𝑑𝑦,𝑥∈ℝ,(2.7) where for each 𝑥,𝑦∈ℝ and 𝑡>0, 𝑊_𝑡(𝑥,𝑦)=𝑒^−𝑡𝜋1−𝑒^−2𝑡^1/2exp−1211+−𝑒𝑒^{−2𝑡−2𝑡}𝑥²+𝑦²+2𝑥𝑦𝑒^−𝑡1−𝑒^−2𝑡(2.8) (see [10]).

Proposition 2.7. Let 𝛼>−1/2, 𝑊_𝑡(𝑥,𝑦) be in (2.8). There exists 𝐶>0 such that, for 𝑡>0, (a)𝑊^𝛼𝑡(𝑥,𝑦)≤𝐶𝑦^𝛼+1/2𝑥^{−𝛼−3/2}, 0<𝑦<𝑥/2, (b)𝑊^𝛼𝑡(𝑥,𝑦)≤𝐶𝑥^𝛼+1/2𝑦^{−𝛼−3/2}, 0<2𝑥<𝑦, (c)|𝑊^𝛼𝑡(𝑥,𝑦)−𝑊_𝑡(𝑥,𝑦)|≤𝐶(1/𝑦), 𝑥/2<𝑦<2𝑥, (d)|𝑊^𝛼𝑡(𝑥,𝑦)|≤𝐶(1/√𝑡)𝑒^{−|𝑥−𝑦|2/10𝑡}.

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<𝑥/2 and 0<2𝑥<𝑦 are not critical; Proposition 2.7 also holds when 0<𝑦<𝑥/𝑐 and 0<𝑐𝑥<𝑦, where 𝑐>1.

Now we consider the estimates of the integral kernel for the 𝑔-function, which will be defined in Section 4: 𝑄_𝑡(𝑥,𝑦)=𝑡²𝜕𝑊^𝛼𝑠(𝑥,𝑦)𝜕𝑠||||_𝑠=𝑡2,(2.9)𝑃_𝑡(𝑥,𝑦)=𝑡²𝜕𝑊_𝑠(𝑥,𝑦)𝜕𝑠||||_𝑠=𝑡2.(2.10)

Proposition 2.9. One has,(a)for every 𝑡,𝑥,𝑦∈(0,∞) such that 𝑒^−𝑡2𝑥𝑦/(1−𝑒^−2𝑡2)≤1, ||𝑄_𝑡(𝑥,𝑦)||≤𝐶𝑡²(𝑥𝑦)^(𝛼+1/2)𝑒^{−(𝑥2+𝑦2)/8𝑡2}𝑒^{−(𝛼+1)𝑡2}1−𝑒^−2𝑡2^𝛼+2,(2.11)(b)for every 𝑡,𝑥,𝑦∈(0,∞) such that 𝑒^−𝑡2𝑥𝑦/(1−𝑒^−2𝑡2)>1, ||𝑄_𝑡(𝑥,𝑦)−𝑃_𝑡(𝑥,𝑦)||≤𝐶𝑡²𝑒^{−(𝑥−𝑦)2/2𝑡2}𝑒^𝑡2/2𝑥𝑦1−𝑒^−𝑡2^1/2.(2.12)

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

Proposition 2.10. For every 𝑁≥1, there is a constant 𝐶_𝑁 such that (a)if 𝑡>0, |𝑃_𝑡(𝑥,𝑦)|≤𝐶_𝑁𝑒^−𝑡2/8(1/𝑡)𝑒^{−|𝑥−𝑦|2/10𝑡2}(1+𝑡|𝑥|)^−𝑁; (b)for |ℎ|≤𝑡, |𝑃_𝑡(𝑥+ℎ,𝑦)−𝑃_𝑡(𝑥,𝑦)|≤𝐶(|ℎ|/𝑡)(1/𝑡)𝑒^{−|𝑥−𝑦|2/20𝑡2}, 𝐶 is independent of 𝑥,𝑦,𝑡; (c)|∫^∞−∞𝑃_𝑡(𝑥,𝑦)𝑑𝑦|≤𝐶(𝑡/𝜌_𝐿𝛼(𝑥))², 𝐶 is independent of 𝑥 and 𝑡.

Proof. By using (2.8) we can write, for every 𝑥,𝑦∈ℝ and 𝑠>0, 𝜕𝜕𝑠𝑊^𝑠(𝑥,𝑦)=−12√𝜋𝑒^{−((𝑥−𝑒−𝑠𝑦)2+(𝑦−𝑒−𝑠𝑥)2)/2(1−𝑒−2𝑠)}𝑒^−𝑠/21−𝑒^−2𝑠^3/2×−12+𝑒𝑒^−2𝑠_−2𝑠+2𝑒^−𝑠(𝑦(𝑥−𝑒^−𝑠𝑦)+𝑥(𝑦−𝑒^−𝑠𝑥))(𝑥−𝑒^−𝑠𝑦)²+(𝑦−𝑒^−𝑠𝑥)²1−𝑒^−2𝑠.(2.13)
By the simple fact (𝑥−𝑒^−𝑠𝑦)²+(𝑦−𝑒^−𝑠𝑥)²=2(𝑥−𝑦)²𝑒^−𝑠+𝑥²+𝑦²(1−𝑒^−𝑠)²,(2.14) a straightforward manipulation leads to |||𝜕𝜕𝑠𝑊^𝑠(𝑥,𝑦)|||≤𝐶𝑒^{−((𝑥−𝑦)2+(𝑥2+𝑦2)(1−𝑒−𝑠)2)/8(1−𝑒−2𝑠)}𝑒^−𝑠/21−𝑒^−2𝑠^3/2≤𝐶_𝑁𝑒^−𝑠/81−𝑒^−2𝑠^3/2𝑒^{−𝑐(|𝑥−𝑦|2/10𝑠)}1+√𝑠|𝑥|^−𝑁,(2.15) which implies (a).
To prove (b), we also directly compute the 𝑥 partial derivative: 𝜕𝜕𝑥𝜕𝑠𝑊^𝑠(𝑥,𝑦)=−12√𝜋𝑒^{−((𝑥−𝑒−𝑠𝑦)2+(𝑦−𝑒−𝑠𝑥)2)/2(1−𝑒−2𝑠)}𝑒^−𝑠/21−𝑒^−2𝑠^3/2×4𝑒^−𝑠(𝑦−𝑒^−𝑠𝑥)−4𝑒^−2𝑠(𝑥−𝑦)+11+𝑒^−𝑠2√𝜋𝑒^{−((𝑥−𝑒−𝑠𝑦)2+(𝑦−𝑒−𝑠𝑥)2)/2(1−𝑒−2𝑠)}𝑒^−𝑠/21−𝑒^−2𝑠^5/2×1+𝑒_−2𝑠((𝑥−𝑒^−𝑠𝑦)−𝑒^−𝑠(𝑦−𝑒^−𝑠𝑥))−2𝑒_−2𝑠+2𝑒^−𝑠(𝑦(𝑥−𝑒^−𝑠𝑦)+𝑥(𝑦−𝑒^−𝑠𝑥))(𝑥−𝑒^−𝑠𝑦)²+(𝑦−𝑒^−𝑠𝑥)²1−𝑒^−2𝑠.(2.16) By an elementary manipulation and (2.14), we have |||𝜕𝜕𝑥𝜕𝑠𝑊^𝑠(𝑥,𝑦)|||≤𝐶1𝑠²𝑒^{−(𝑥−𝑦)2/16𝑠}.(2.17) This together with the mean value theorem and the condition |ℎ|≤𝑡 leads to (b).
Let 𝜙_𝑛(𝑦)=𝜙(𝑦/𝑛); 𝜙(𝑦) is a smooth function satisfying 𝜙(𝑦)=1 for |𝑦|≤1, 𝜙(𝑦)=0 for |𝑦|≥2 and Δ𝜙(𝑦)≤1 for 𝑦∈ℝ. From the above, for fixed 𝑠 and 𝑥, a straightforward manipulation shows that ^+∞−∞||||𝜕𝑊_𝑠(𝑥,𝑦)𝜕𝑠||||𝑑𝑦<∞.(2.18) Hence, we have ||||^∞−∞𝜕𝑊_𝑠(𝑥,𝑦)𝜕𝑠𝑑𝑦||||=||||lim^𝑛→∞^∞−∞𝜕𝑊_𝑠(𝑥,𝑦)𝜕𝑠𝜙^𝑛=||||lim_𝑛→∞(𝑦)𝑑𝑦||||^∞−∞≤𝐶_∞−∞𝑊_𝑠(𝑥,𝑦)𝐻𝜙_𝑛(𝑦)𝑑𝑦||||𝑊_𝑠(𝑥,𝑦)𝑦²𝑑𝑦.(2.19) Using (2.8) again, 𝐼=||||^∞−∞𝜕𝑊_𝑠(𝑥,𝑦)≤𝐶_∞−∞𝜕𝑠𝑑𝑦||||𝑒^−𝑠/4√𝑠exp−(𝑥−𝑦)²𝑒^−𝑠+𝑥²+𝑦²(1−𝑒^−𝑠)²21−𝑒^−2𝑠(𝑦−𝑥)²+𝑥²𝑑𝑦,(2.20) which implies (c).

Lemma 2.11 (see [3, Theorem 2]). For all 𝑓∈BMO_𝐿𝛼 and 𝐵=𝐵_𝑟(𝑦₀)⊆(0,∞), there exists a constant 𝐶>0 such that 1||𝐵||^𝑟0_𝐵𝑄^2𝑡𝑓(𝑥)𝑑𝑥𝑑𝑡𝑡≤𝐶‖𝑓‖^2BMO𝐿𝛼.(2.21)

3. Maximal Operators

First of all, we define the following notions: 𝑀₊𝑓(𝑥)=sup_{𝑥∈𝐵⊆(0,∞)}1||𝐵||_𝐵ℐ||𝑓(𝑦)||𝑑𝑦,(3.1)_∗𝛼𝑓(𝑥)=sup_𝑡>0||𝑇^𝛼𝑡𝑓(𝑥)||.(3.2)

In this section, we will show ℐ^∗𝛼 and 𝑀₊ are bounded on BMO_𝐿𝛼.

Theorem 3.1. There exists a constant 𝐶>0 such that, for all 𝑓∈BMO_𝐿𝛼, 𝑀₊𝑓<∞, for a.e. 𝑥∈(0,∞), and ‖‖𝑀₊𝑓‖‖_BMO𝐿𝛼≤𝐶‖𝑓‖_BMO𝐿𝛼.(3.3)

Proof. First of all, we show that for a.e. 𝑥∈(0,∞), 𝑀₊𝑓<∞. To do this, we only need to show that for, at almost 𝑥∈𝐵_𝑘⊆ℬ in Lemma 2.1, 𝑀₊𝑓(𝑥)<∞. Let us split 𝑓=𝑓₁+𝑓₂ with 𝑓₁=𝑓𝜒_𝐵^∗𝑘. Obviously, since 𝑓 is locally integrable, we have 𝑀₊𝑓₁<∞ for a.e. 𝑥∈(0,∞). For 𝑓₂, if 𝑥∈𝐵 and 𝐵∩𝐵^∗𝑘=∅, since supp 𝑓₂ is in the complement of 𝐵^∗𝑘, we have (1/|𝐵|)∫_𝐵|𝑓(𝑦)|𝑑𝑦=0. Otherwise, by the definition of BMO_𝐿𝛼, (1/|𝐵|)∫_𝐵|𝑓(𝑦)|𝑑𝑦≤(4/|𝐵_4𝑟(𝑥_𝑘)|)∫_{𝐵4𝑟(𝑥𝑘)}|𝑓(𝑦)|𝑑𝑦≤𝑐‖𝑓‖_BMO𝐿𝛼.
We turn to the boundedness in BMO_𝐿𝛼. 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 (−∞,0]. Notice that 𝑀₊𝑓=𝑀𝑓₀, for 𝑥∈(0,∞), so ‖‖𝑀₊𝑓‖‖_{BMO(𝐵∗𝑘)}=‖‖𝑀𝑓₀‖‖_{BMO(𝐵∗𝑘)}≤𝐶‖‖𝑓₀‖‖_BMO.(3.4) Now, we need to show that ‖𝑓₀‖_BMO≤𝐶‖𝑓‖_BMO𝐿𝛼. Indeed, if 𝐵⊆(0,∞), it is obvious that (1/|𝐵|)∫_𝐵|𝑓₀−(𝑓₀)_𝐵|𝑑𝑦≤‖𝑓‖_BMO𝐿𝛼. If 𝐵∩(0,∞)=∅, then (1/|𝐵|)∫_𝐵|𝑓₀−(𝑓₀)_𝐵|𝑑𝑦=0. If 𝐵∩(0,∞)≠∅ and 𝐵∩(−∞,0)≠∅, let 𝐵=𝐵₁∪𝐵₂, here 𝐵₁=𝐵∩(−∞,0) and 𝐵₂=𝐵∩(0,∞), then 1||𝐵||_𝐵||𝑓₀−𝑓₀_𝐵||𝑑𝑦≤21||𝐵||_𝐵2≤2‖𝑓‖||𝑓_0BMO𝐿𝛼||𝑑𝑦.(3.5) On the other hand, we again split 𝑓=𝑓₁+𝑓₂ with 𝑓₁=𝑓𝜒_𝐵^∗𝑘, from the argument above, 𝑀₊𝑓₂(𝑥)≤𝑐‖𝑓‖_BMO𝐿𝛼, for a.e 𝑥∈𝐵_𝑘. So 1||𝐵_𝑘||_𝐵𝑘||𝑀₊𝑓||𝑑𝑦≤1||𝐵_𝑘||_𝐵𝑘||𝑀₊𝑓₁||𝑑𝑦+1||𝐵_𝑘||_𝐵𝑘≲1||𝑀₊𝑓₂||𝑑𝑦||𝐵_𝑘||_𝐵𝑘||𝑀₊𝑓₁||²𝑑𝑦^1/2+‖𝑓‖_BMO𝐿𝛼≲‖𝑓‖_BMO𝐿𝛼,(3.6) where in the last inequality we have used Corollary 2.4.

Theorem 3.2. Let 𝛼>−1/2. There exists a constant 𝐶>0 such that ‖‖ℐ^∗𝛼𝑓‖‖_BMO𝐿𝛼≤𝐶‖𝑓‖_BMO𝐿𝛼.(3.7)

Proof. By the definition of BMO_𝐿𝛼 and Corollary 2.3, it suffices to prove the following: for every fixed “critical ball” 𝐵_𝑘∈ℬ (see Lemma 2.1) we have (1)(1/|𝐵_𝑘|)∫_𝐵𝑘|ℐ^∗𝛼𝑓|𝑑𝑥≤𝐶‖𝑓‖_BMO𝐿𝛼, (2)‖ℐ^∗𝛼𝑓‖_BMO(𝐵^∗𝑘₎≤𝐶‖𝑓‖_BMOL𝛼.
Let us start to prove (1). It is immediate from Theorem 3.1 and (d) of Proposition 2.7; since ℐ^∗𝛼𝑓(𝑥)≤𝑀₊𝑓(𝑥), for 𝑥>0, therefore, 1||𝐵_𝑘||_𝐵𝑘||𝑀₊𝑓||𝑑𝑦≤𝐶‖𝑓‖_BMO𝐿𝛼.(3.8)
It remains to show (2). By Lemma 2.5, we split ℐ^∗𝛼𝑓(𝑥) into several parts. First, we shall show ‖‖‖‖‖sup_{𝑡>𝜌2𝐿𝛼𝑥𝑘}||𝑇^𝛼𝑡𝑓(𝑥)||‖‖‖‖‖_{𝐿∞(𝐵∗𝑘)}≤𝐶‖𝑓‖_BMO𝐿𝛼.(3.9)
From (d) of Proposition 2.7, we have ||𝑇^𝛼𝑡𝑓(𝑥)||≲^∞0||𝑓(𝑦)||1𝑡^1/21+||𝑥√−𝑡𝑦||^−𝑁≲𝑑𝑦_∞_𝑗=012^𝑗𝑁1𝑡^1/2_{{𝑦>0,|𝑦−𝑥|<2𝑗√𝑡}}||𝑓(𝑦)||𝑑𝑦.(3.10)
Notice that, for 𝑗≥0 and 𝑡>𝜌^2𝐿_{𝛼(𝑥𝑘)}, we have 2^𝑗√𝑡≥𝜌_{𝐿𝛼(𝑥)}∼𝜌_{𝐿𝛼(𝑥𝑘)}, for 𝑥∈𝐵^∗𝑘. Thus 1√𝑡^{{𝑦>0,|𝑦−𝑥|<2𝑗√𝑡}}||𝑓(𝑦)||𝑑𝑦≤𝐶2^𝑗‖𝑓‖_BMO𝐿𝛼.(3.11)
Therefore, sup_{𝑡>𝜌2𝐿𝛼𝑥𝑘}||𝑇^𝛼𝑡𝑓(𝑥)||≲^∞_𝑗=012^𝑁−1‖𝑓‖_BMO𝐿𝛼≲‖𝑓‖_BMO𝐿𝛼.(3.12)
By Lemma 2.5, it suffices to show that sup_{0<𝑡≤𝜌2𝐿𝛼𝑘)(𝑥}|𝑇^𝛼𝑡𝑓(𝑥)| satisfies (2). Write 𝑓=𝑓𝜒_{{𝑥𝑘/2≤𝑦≤2𝑥𝑘}}+𝑓𝜒_{{𝑦<𝑥𝑘/2}}+𝑓𝜒_{{𝑦>2𝑥𝑘}}=𝑓₁+𝑓₂+𝑓₃.(3.13)
By Proposition 2.7, it easily follows that ‖‖‖‖‖sup_{0<𝑡≤𝜌2𝐿𝛼𝑥𝑘}‖‖‖‖‖sup_{0<𝑡≤𝜌2𝐿𝛼𝑥𝑘}||𝑇^𝛼𝑡𝑓₂(𝑥)||‖‖‖‖‖_{𝐿∞(𝐵∗𝑘)}≤𝐶‖𝑓‖_BMO𝐿𝛼,||𝑇^𝛼𝑡𝑓₃(𝑥)||‖‖‖‖‖_{𝐿∞(𝐵∗𝑘)}≤𝐶‖𝑓‖_BMO𝐿𝛼.(3.14)
Indeed, since 𝑥∼𝑥_𝑘, for 𝑥∈𝐵^∗𝑘, by (a) of Proposition 2.7 and Remark 2.8, we have ||𝑇^𝛼𝑡𝑓₂(𝑥)||≲≲1^𝑥𝑘0/2𝑦^𝛼+1/2𝑥^{𝑘−𝛼−3/2}||𝑓(𝑦)||𝑑𝑦𝑥_𝑘^𝑥𝑘0/2≲‖𝑓‖||𝑓_BMO𝐿𝛼(𝑦)||𝑑𝑦.(3.15)
Similarly, ||𝑇^𝛼𝑡𝑓₃(𝑥)||≲^∞_𝑛=12^𝑛𝑥_𝑘^{−𝛼−3/2}𝑥_𝑘^𝛼+1/2^{2𝑛+1𝑥𝑘}_{2𝑛𝑥𝑘}≲||𝑓(𝑦)||𝑑𝑦_∞_𝑛=1(2^𝑛)^{−𝛼−1/2}‖𝑓‖_BMO𝐿𝛼≲‖𝑓‖_BMO𝐿𝛼.(3.16)
Now, we come to treat 𝑓₁. We make further decompositions. Split 𝑇^𝛼𝑡𝑓₁=𝑇^𝛼𝑡𝑓₁−𝑊_𝑡𝑓₁+𝑊_𝑡𝑓₁−𝐻_𝑡𝑓₁+𝐻_𝑡𝑓₁,(3.17)
where 𝐻_𝑡𝑔(𝑥)=^∞0𝜋1𝑒−^−𝑡𝑒^−2𝑡^1/2exp−1211+−𝑒𝑒^{−2𝑡−2𝑡}(𝑥−𝑦)²𝑔(𝑦)𝑑𝑦.(3.18)
For the first term, by (c) of Proposition 2.7, we have ||𝑇^𝛼𝑡𝑓₁−𝑊_𝑡𝑓₁||≤𝐶1𝑥_𝑘^2𝑥𝑘_𝑥2/2||𝑓(𝑦)||𝑑𝑦≤𝐶‖𝑓‖_BMO𝐿𝛼.(3.19)
By (2.8), 𝑊_𝑡𝑓₁=(𝑥)−𝐻_𝑡𝑓₁(𝑥)_∞0𝜋1𝑒−^−𝑡𝑒^−2𝑡^1/2exp−1211+−𝑒𝑒^{−2𝑡−2𝑡}(𝑥−𝑦)²𝑒^{−2𝑥𝑦(1−𝑒−𝑡)2/1−𝑒−2𝑡}−1𝑓₁(𝑦)𝑑𝑦.(3.20)
Notice that |𝑒^{−2𝑥𝑦(1−𝑒−𝑡)2/(1−𝑒−2𝑡)}−1|≤𝑐𝑡𝑥^2𝑘, when 𝑡≤𝜌_𝐿𝛼(𝑥_𝑘)², 𝑥_𝑘/2≤𝑦≤2𝑥_𝑘 and 𝑥∈𝐵^∗𝑘. Therefore, for 𝑥∈𝐵^∗𝑘 and 𝑡≤𝜌_𝐿𝛼(𝑥_𝑘)², we obtain ||𝑊_𝑡𝑓₁(𝑥)−𝐻_𝑡𝑓₁(𝑥)||≲𝑡𝑥^2𝑘^∞01√𝑡𝑒^{−𝑐0|𝑥−𝑦|2/𝑡}≲𝑡𝑥_2𝑘||𝑓₁(𝑦)||𝑑𝑦_{1≤2𝑗≤𝜌}_{𝐿𝛼𝑥𝑘/√𝑡}2^{−𝑗(𝑁−1)}12^𝑗√𝑡^{{𝑦>0,|𝑦−𝑥|<2𝑗√𝑡}}+𝑡𝑥_2𝑘||𝑓(𝑦)||𝑑𝑦_{𝜌𝐿𝛼𝑥}_{𝑘/√𝑡<2𝑗}2^{−𝑗(𝑁−1)}12^𝑗√𝑡^{{𝑦>0,|𝑦−𝑥|<2𝑗√𝑡}}≲||𝑓(𝑦)||𝑑𝑦_{1≤2𝑗≤𝜌𝐿𝛼𝑥𝑘/√𝑡}2^{−𝑗(𝑁−1)}𝑡𝑥^2𝑘1+log𝜌^𝐿𝛼𝑥_𝑘2^𝑗√𝑡‖𝑓‖^BMO𝐿𝛼+^∞_𝑗=02^{−𝑗(𝑁−1)}‖𝑓‖_BMO𝐿𝛼≲^∞_𝑗=02^{−𝑗(𝑁−1)}‖𝑓‖_BMO𝐿𝛼≲‖𝑓‖_BMO𝐿𝛼.(3.21)
Finally, by Lemma 2.5 again, we need to show that sup_{0<𝑡≤𝜌2𝐿𝛼𝑘)(𝑥}|𝐻_𝑡𝑓₁(𝑥)| satisfies (2). Consider 𝐵=𝐵_𝑟(𝑥₀)⊂𝐵^∗𝑘 and write 𝑓₁==𝑓𝑓₁₁₁−𝑓_𝐵^∗𝜒_𝐵^∗+𝑓₁−𝑓_𝐵^∗𝜒_(𝐵^∗₎^𝑐_∩(0,∞)+𝑓_𝐵^∗𝜒_(0,∞)+𝑓₁₂.(3.22) By Corollary 2.3, we choose a constant 𝐶_𝐵=sup_{0<𝑡≤𝜌𝐿𝛼(𝑥𝑘)}²|𝐻_𝑡𝑓₁₂(𝑥₀)|, 1||𝐵||_𝐵|||||sup_{0<𝑡≤𝜌𝐿𝛼𝑥𝑘2}||𝐻_𝑡𝑓₁(𝑥)||−𝐶_𝐵|||||𝑑𝑥≤1||𝐵||_𝐵≤1sup_{0<𝑡≤𝜌𝐿𝛼𝑥𝑘2}||𝐻_𝑡𝑓₁(𝑥)−𝐻_𝑡𝑓₁₂𝑥₀||𝑑𝑥||𝐵||_𝐵sup_{0<𝑡≤𝜌𝐿𝛼𝑥𝑘2}||𝐻_𝑡𝑓₁₁(𝑥)||𝑑𝑥+1||𝐵||_𝐵sup_{0<𝑡≤𝜌𝐿𝛼𝑥𝑘2}||𝐻_𝑡𝑓₁₂(𝑥)−𝐻_𝑡𝑓₁₂𝑥₀||𝑑𝑥.(3.23) For the first integral, by Corollary 2.4 it easily follows that 1||𝐵||_𝐵sup_{0<𝑡≤𝜌𝐿𝛼𝑥𝑘2}||𝐻_𝑡𝑓₁₁(𝑥)||𝑑𝑥≤1||𝐵||_𝐵||𝑓₁₁||²𝑑𝑥^1/2≤𝐶‖𝑓‖_BMO𝐿𝛼.(3.24) For the second integral, ||𝐻_𝑡𝑓≤₁(𝑥||||)−𝐻_𝑡𝑓_{12(𝐵∗)𝑐∩(0,∞)}𝑥₀||=𝐼_𝑡1𝐻_𝑡(𝑥,𝑦)−𝐻_𝑡𝑥₀,𝑦𝑓₁(𝑦)−𝑓_𝐵𝑑𝑦||||+||||^∞0𝐻_𝑡(𝑥,𝑦)−𝐻_𝑡𝑥₀,𝑦𝑓_𝐵𝑑𝑦||||(𝑥)+𝐼^𝑡2(𝑥).(3.25) By the mean value theorem and the elementary inequality 12^𝑛𝑟_{{𝑦>0,|𝑦−𝑥0|<2}^𝑛_𝑟}||𝑓(𝑦)−𝑓_𝐵||𝑑𝑦≤𝑐𝑛‖𝑓‖_BMO𝐿𝛼,(3.26) we have 𝐼^𝑡1(𝑥)≲_{(𝐵∗)𝑐∩(0,∞)}1√𝑡𝑒^{−|𝑦−𝑥0|2/10𝑡}||𝑥−𝑥₀||√𝑡||𝑓¹≲(𝑦)−𝑓_𝐵||𝑑𝑦_{(𝐵∗)𝑐∩(0,∞)}||𝑥−𝑥₀||||𝑦−𝑥₀||²≲𝑟_∞||𝑓₁(𝑦)−𝑓_𝐵||𝑑𝑦_𝑛=01(2^𝑛𝑟)²_{{𝑦>0,|𝑦−𝑥0|<2𝑛𝑟}}≲||𝑓(𝑦)−𝑓_𝐵||+||𝑓(𝑦)−𝑓₁(𝑦)||𝑑𝑦_∞_𝑛=02^−𝑛(𝑛+1)‖𝑓‖_BMO𝐿𝛼.(3.27) On the other hand, by the fact |𝑓_𝐵|≤𝐶(1+log(𝜌_𝐿𝛼(𝑥₀)/𝑟))‖𝑓‖_BMO𝐿𝛼 in Lemma 2.6, we obtain 𝐼^𝑡2(𝑥)≲≲||||^0−∞𝐻_𝑡(𝑥,𝑦)−𝐻_𝑡𝑥₀,𝑦𝑓_𝐵𝑑𝑦_{𝑦<0}||||||𝑥−𝑥₀||||𝑦−𝑥₀||²≲𝑟||𝑓_𝐵||𝑑𝑦𝜌_𝐿𝛼𝑥₀1+log𝜌^𝐿𝛼𝑥₀𝑟‖𝑓‖^BMO𝐿𝛼≲‖𝑓‖_BMO𝐿𝛼.(3.28) Therefore, we obtain 1||𝐵||_𝐵sup_{0<𝑡≤𝜌𝐿𝛼𝑥𝑘2}||𝐻_𝑡𝑓₁₂(𝑥)−𝐻_𝑡𝑓₁₂𝑥₀||𝑑𝑥≲‖𝑓‖_BMO𝐿𝛼,(3.29) which establishes the proof.

4. 𝑔-Function

For all 𝑓∈𝐿^1loc(0,∞) and 𝑥∈(0,∞), define the Littlewood-Paley 𝑔-function by 𝑔(𝑓)(𝑥)≡^∞0||𝑄_𝑡𝑓(𝑥)||²𝑑𝑡𝑡^1/2,(4.1) where, {𝑄_𝑡}_𝑡>0 is a family of operators with the integral kernels 𝑄_𝑡(𝑥,𝑦)=𝑡²𝜕𝑊^𝛼𝑠(𝑥,𝑦)𝜕𝑠||||_𝑠=𝑡2.(4.2)

Theorem 4.1. Let 𝛼>−1/2. There exists a constant 𝐶>0 such that, for all 𝑓∈BMO_𝐿𝛼, 𝑔(𝑓)∈BMO_𝐿𝛼 and ‖𝑔(𝑓)‖_BMO𝐿𝛼≤𝐶‖𝑓‖_BMO𝐿𝛼.

Proof. By Proposition 2.9 and (a) of Proposition 2.10, we have 𝑄_𝑡(𝑥,𝑦)≤𝑐1𝑡𝑒^{−𝑐1|𝑥−𝑦|2/𝑡2}.(4.3) For 𝑓∈BMO_𝐿𝛼, because of this and the integrability of (1+|𝑥|)⁻²𝑓(𝑥) (see [12, page 141]), 𝑄_𝑡𝑓(𝑥)=^∞0𝑄_𝑡(𝑥,𝑦)𝑓(𝑦)𝑑𝑦(4.4) is well defined absolutely convergent integral for all (𝑥,𝑡)∈(0,∞)×(0,∞). Similar to the proof of Theorem 3.2, we will try to show that, for 𝐵_𝑘⊂ℬ in Lemma 2.1, (1)(1/|𝐵_𝑘|)∫_𝐵𝑘|𝑔(𝑓)(𝑥)|𝑑𝑥≤𝐶‖𝑓‖_BMO𝐿𝛼, (2)‖𝑔(𝑓)(𝑥)‖_BMO(𝐵^∗𝑘₎≤𝐶‖𝑓‖_BMO𝐿𝛼. We split [𝑔(𝑓)(𝑥)]²=𝑔₁(𝑓)(𝑥)²+𝑔₂(𝑓)(𝑥)²=^{20𝜌𝐿𝛼(𝑥𝑘)0}||𝑄_𝑡𝑓(𝑥)||²𝑑𝑡𝑡+^∞20𝜌_{𝐿𝛼𝑥𝑘}||𝑄_𝑡𝑓(𝑥)||²𝑑𝑡𝑡.(4.5) By Lemma 2.11 and Hölder inequality, assertion (1) holds for 𝑔₁(𝑓)(𝑥). To finish the proof of (1), it suffices to show that ‖‖𝑔₂(𝑓)‖‖_{𝐿∞(𝐵∗𝑘)}≤𝑐‖𝑓‖_BMO𝐿𝛼.(4.6)
In the next proof, for the sake of brevity we introduce the additional notations: 𝑋^𝑡1(𝑥)=𝑦∈(0,∞)∶𝑒^−𝑡2𝑥𝑦1−𝑒^−2𝑡2𝑋≤1,(4.7)_𝑡2(𝑥)=&𝑦∈(0,∞)∶𝑒^−𝑡2𝑥𝑦1−𝑒^−2𝑡2>1.(4.8) By 𝑋^𝑡1(𝑥) and 𝑋^𝑡1(𝑥), we split 𝑄_𝑡𝑓(𝑥) as ||𝑄_𝑡𝑓(𝑥)||≤+_{𝑋𝑡1(𝑥)}||𝑓(𝑦)||||𝑄_𝑡(𝑥,𝑦)||𝑑𝑦+_{𝑋𝑡2(𝑥)}||𝑓(𝑦)||||𝑄_𝑡(𝑥,𝑦)−𝑃_𝑡(𝑥,𝑦)||𝑑𝑦_{𝑋𝑡2(𝑥)}=𝐼_𝑡1||𝑓(𝑦)||||𝑃_𝑡(𝑥,𝑦)||𝑑𝑦(𝑥)+𝐼^𝑡2(𝑥)+𝐼^𝑡3(𝑥).(4.9)
For 𝐼^𝑡1(𝑥) and 𝑥∈𝐵^∗𝑘, we shall first show the inequality 𝐽₁(𝑥)=^∞20𝜌_{𝐿𝛼𝑥𝑘}||𝐼^𝑡1𝑓(𝑥)||²𝑑𝑡𝑡≤𝑐‖𝑓‖^2BMO𝐿𝛼.(4.10) Using (a) of Proposition 2.9, if 𝑥_𝑘≤1, 𝜌_𝐿𝛼(𝑥_𝑘)∼𝑥_𝑘, we get ||𝑄_𝑡(𝑥,𝑦)||≤𝐶𝑡^2𝛼+3𝑥𝑡²𝑦^𝛼+1/2𝑒^{−(𝑥2+𝑦2)/8𝑡2}𝑒^{−(𝛼+1)𝑡2}1−𝑒^−2𝑡2^𝛼+2≤𝐶𝑥^𝑘𝑡^𝛼+1/21𝑡𝑒^{−|𝑥−𝑦|2/16𝑡2}.(4.11)
If 𝑥_𝑘≥1, 𝜌_𝐿𝛼(𝑥_𝑘)∼1/𝑥_𝑘, we have ||𝑄_𝑡(𝑥,𝑦)||≤𝐶1𝑡𝑒^{−(𝑥2+𝑦2)/8𝑡2}𝑒^{−((𝛼+1)/2)𝑡2}≤𝐶1𝑡𝑥_𝑘1𝑡𝑒^{−|𝑥−𝑦|2/16𝑡2}.(4.12) The previous two inequalities above imply 𝐽₁(𝑥)≲^∞20𝜌_{𝐿𝛼𝑥𝑘}|||||_∞0𝜌^𝐿𝛼𝑥_𝑘𝑡^𝜎1𝑡𝑒^{−|𝑥−𝑦|2/16𝑡2}||𝑓(𝑦)||𝑑𝑦|||||²𝑑𝑡≲_{∞20𝜌𝐿𝛼𝑥𝑘}𝑡𝜌^𝐿𝛼𝑥_𝑘𝑡^2𝜎|||||_∞_𝑗=02^{−𝑗(𝑁−1)}12^𝑗𝑡_{{𝑦>0,|𝑦−𝑥|<2}^𝑗_𝑡}||𝑓(𝑦)||𝑑𝑦|||||²𝑑𝑡𝑡≲‖𝑓‖^2BMO𝐿𝛼.(4.13)
For 𝐼^𝑡2(𝑥) and 𝑥∈𝐵^∗𝑘, we shall also prove the inequality 𝐽₂(𝑥)=^∞20𝜌_{𝐿𝛼𝑥𝑘}||𝐼^𝑡2𝑓(𝑥)||²𝑑𝑡𝑡≤𝑐‖𝑓‖^2BMO𝐿𝛼.(4.14) We split this integral as 𝐽₂(𝑥)=^120𝜌_{𝐿𝛼𝑥𝑘}||𝐼^𝑡2𝑓(𝑥)||²𝑑𝑡𝑡+^∞1||𝐼^𝑡2𝑓(𝑥)||²𝑑𝑡=𝐽₃𝑡(𝑥)+𝐽₄(𝑥).(4.15) To deal with 𝐽₃(𝑥), we discuss two cases. In the first case of 𝑥_𝑘≤1, notice that 𝑦>𝑥, when 𝑥∈𝐵^∗𝑘, 𝑦∈𝑋^𝑡2(𝑥_𝑘) and 𝑡≥20𝜌_𝐿𝛼(𝑥_𝑘). According to (b) of Proposition 2.9, 𝐽₃(𝑥)≲^120𝜌_{𝐿𝛼𝑥𝑘}|||||_{∞1−𝑒2−2𝑡𝑒𝑡2/𝑥}1𝑡𝑒^{−|𝑥−𝑦|2/4𝑡2}𝑥𝑡1−𝑒^−2𝑡2𝑒^𝑡2−𝑥²||𝑓(𝑦)||𝑑𝑦|||||²𝑑𝑡≲_{∞20𝜌𝐿𝛼𝑥𝑘}𝑡𝑥^2𝑘𝑡²||||^∞01𝑡𝑒^{−|𝑥−𝑦|2/4𝑡2}||𝑓(𝑦)||𝑑𝑦||||²𝑑𝑡≲‖𝑓‖_2BMO𝐿𝛼𝑡.(4.16) The last inequality is from the same proof of 𝐽₁(𝑥). In the second case of 𝑥_𝑘>1, using (b) of Proposition 2.9 again, for 𝑡>20𝜌_𝐿𝛼(𝑥_𝑘) we obtain ||𝐼^𝑡2(𝑥)||≲1𝑥_𝑘^2𝑥𝑘_𝑥𝑘/2𝑡𝑥_𝑘||𝑓(𝑦)||𝑑𝑦+_{(0,∞)∩(𝑥𝑘/2,2𝑥𝑘)𝑐}1𝑡𝑒^{−|𝑥−𝑦|2/2𝑡2}≲𝑡||𝑓(𝑦)||𝑑𝑦𝑥_𝑘‖𝑓‖_BMO𝐿𝛼+𝑡𝑥_𝑘^∞01𝑡𝑒^{−|𝑥−𝑦|2/4𝑡2}≲𝑡||𝑓(𝑦)||𝑑𝑦𝑥_𝑘‖𝑓‖_BMO𝐿𝛼.(4.17) The last inequality is also from the same proof of 𝐽₁(𝑥). Inserting this into 𝐽₃(𝑥) leads to 𝐽₃(𝑥)≤𝐶‖𝑓‖²_BMO𝐿𝛼. Now, it remains to show 𝐽₄(𝑥)≤𝑐‖𝑓‖²_BMO𝐿𝛼. Using (b) of Proposition 2.9, by the standard argument it easily follows that 𝐽₄(𝑥)≲^∞1𝑒^−𝑡/10𝑡||||^∞01𝑡𝑒^{−|𝑥−𝑦|2/2𝑡2}||𝑓(𝑦)||𝑑𝑦||||²𝑑𝑡≲‖𝑓‖²_BMO𝐿𝛼.(4.18) To complete the proof of (4.6), we need to show that 𝐽₅(𝑥)=^∞20𝜌_{𝐿𝛼𝑥𝑘}||𝐼^𝑡3𝑓(𝑥)||²𝑑𝑡𝑡≤𝑐‖𝑓‖^2BMO𝐿𝛼.(4.19) We also consider two cases of 𝑥_𝑘≤1 and 𝑥_𝑘>1. When 𝑥_𝑘≤1, repeating the above argument for 𝐽₃(𝑥) and using (a) of Proposition 2.10, we have 𝐽₅≤𝑐‖𝑓‖²_BMO𝐿𝛼. When 𝑥_𝑘>1, using (a) of Proposition 2.10 again, for 𝑡≥20𝜌_𝐿𝛼(𝑥_𝑘), we obtain ||𝐼^𝑡3(𝑥)||≲1𝑥_𝑘𝑡^∞01𝑡𝑒^{−𝑐|𝑦−𝑥|2/𝑡2}≲1||𝑓(𝑦)||𝑑𝑦𝑡𝑥_{𝑘∞𝑗=0}2^{−𝑗(𝑁−1)}12^𝑗𝑡_{{𝑦>0,|𝑦−𝑥|<2}^𝑗_𝑡}≲𝜌||𝑓(𝑦)||𝑑𝑦_𝐿𝛼𝑥_𝑘𝑡‖𝑓‖^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 ‖‖‖‖^20𝜌_𝐿𝛼^(𝑥_𝑘⁾⁰||𝑄_𝑡𝑓(𝑥)||²𝑑𝑡𝑡^1/2‖‖‖‖_{BMO(𝐵∗𝑘)}≤𝐶‖𝑓‖_BMO𝐿𝛼.(4.21)
Consider any ball 𝐵=𝐵_𝑟(𝑥₀)⊂𝐵^∗𝑘. By Lemma 2.11, we have 1||𝐵||_𝐵^𝑟0||𝑄_𝑡𝑓(𝑥)||²𝑑𝑡𝑡^1/2𝑑𝑥≤1||𝐵||_𝐵^𝑟0||𝑄_𝑡𝑓(𝑥)||²𝑑𝑡𝑡𝑑𝑥^1/2≤𝐶‖𝑓‖_BMO𝐿𝛼.(4.22) Therefore, by Lemma 2.5 and Corollary 2.3, it suffices to prove 1||𝐵||_𝐵|||||_{20𝜌𝐿𝛼(𝑥𝑘)𝑟}||𝑄_𝑡𝑓(𝑥)||²𝑑𝑡𝑡^1/2−^{20𝜌𝐿𝛼(𝑥𝑘)𝑟}||𝑄_𝑡𝑓𝑥₀||²𝑑𝑡𝑡^1/2|||||𝑑𝑥≤𝐶‖𝑓‖_BMO𝐿𝛼.(4.23) To prove (4.23), we first claim that, for all 𝑓∈BMO_𝐿𝛼, 𝑥∈𝐵^∗𝑘, and 𝑡≤20𝜌_𝐿𝛼(𝑥_𝑘), ||𝑄_𝑡𝑓(𝑥)||≤𝐶‖𝑓‖_BMO𝐿𝛼.(4.24)
We shall split into three different estimates: _{𝑋𝑡1(𝑥)}||𝑓(𝑦)||||𝑄_𝑡(𝑥,𝑦)||𝑑𝑦≤𝐶‖𝑓‖_BMO𝐿𝛼,(4.25)_{𝑋𝑡2(𝑥)}||||||𝑓(𝑦)||||𝑄_𝑡(𝑥,𝑦)−𝑃_𝑡(𝑥,𝑦)||𝑑𝑦≤𝐶‖𝑓‖_BMO𝐿𝛼,(4.26)_{𝑋𝑡2(𝑥)}𝑓(𝑦)𝑃_𝑡(𝑥,𝑦)𝑑𝑦||||≤𝐶‖𝑓‖_BMO𝐿𝛼.(4.27) Let us first treat (4.25). Since 𝑦≤𝑐(𝜌_𝐿𝛼(𝑥_𝑘)²/𝑥_𝑘), when 𝑦∈𝑋^𝑡1(𝑥), notice that 𝑥∼𝑥_𝑘 when 𝑥∈𝐵^∗𝑘, using (a) of Proposition 2.9, and recalling the definition of 𝜌_𝐿𝛼(𝑥), we have _{𝑋𝑡1(𝑥)}||𝑓(𝑦)||||𝑄_𝑡(𝑥,𝑦)||𝑑𝑦≤𝐶^{𝑐𝜌𝐿𝛼(𝑥𝑘)2/𝑥𝑘}₀𝑡𝑥^2𝑘||𝑓(𝑦)||𝑑𝑦≤𝐶𝑡𝑥_𝑘‖𝑓‖_BMO𝐿𝛼≤𝐶‖𝑓‖_BMO𝐿𝛼.(4.28) For (4.26), using (b) of Proposition 2.9, for 𝑥∈𝐵^∗𝑘, the left side of (4.26) is controlled by 1𝑥_𝑘^2𝑥𝑘_𝑥𝑘/2𝑡𝑥_𝑘||𝑓(𝑦)||𝑑𝑦+_{(0,∞)∩(𝑥𝑘/2,2𝑥𝑘)𝑐}1𝑡𝑒^{−|𝑥−𝑦|2/2𝑡2}≲𝑡||𝑓(𝑦)||𝑑𝑦𝑥_𝑘‖𝑓‖_BMO𝐿𝛼+𝑡𝑥_𝑘^∞01𝑡𝑒^{−|𝑥−𝑦|2/4𝑡2}≲𝑡||𝑓(𝑦)||𝑑𝑦𝑥_𝑘‖𝑓‖_BMO𝐿𝛼+𝑡𝑥_𝑘12‖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 ||||_{𝑋𝑡2(𝑥)}𝑓(𝑦)𝑃_𝑡(𝑥,𝑦)𝑑𝑦||||≤||||_{𝑋𝑡2(𝑥)}||𝑓(𝑦)−𝑓_{𝐵𝑡(𝑥)}||||𝑃_𝑡(𝑥,𝑦)||𝑑𝑦||||+||||_{𝑋𝑡2(𝑥)}𝑓_{𝐵𝑡(𝑥)}𝑃_𝑡(𝑥,𝑦)𝑑𝑦||||.(4.30)
By (a), (c) of Proposition 2.10 and the fact that |𝑓_{𝐵𝑡(𝑥)}|≲(1+log20𝜌_𝐿𝛼(𝑥_𝑘)/𝑡)‖𝑓‖_BMO𝐿𝛼, we have ||||_{𝑋𝑡2(𝑥)}𝑓_{𝐵𝑡(𝑥)}𝑃_𝑡(𝑥,𝑦)𝑑𝑦||||≤≲||||^∞−∞𝑡𝑓_{𝐵𝑡(𝑥)}𝑃_𝑡(𝑥,𝑦)𝑑𝑦||||+|||||^{𝑒𝑡2(1−𝑒2−2𝑡)/𝑥−∞}𝑓_{𝐵𝑡(𝑥)}𝑃_𝑡(𝑥,𝑦)𝑑𝑦₂|||||𝜌_𝐿𝛼𝑥_𝑘²1+log20𝜌^𝐿𝛼𝑥_𝑘𝑡‖𝑓‖^BMO𝐿𝛼+^{(1−𝑒2−2𝑡)𝑒𝑡2/𝑥−∞}1𝑡𝑒^{−(𝑥−𝑦)2/16𝑡2}𝑑𝑦1+log20𝜌^𝐿𝛼𝑥_𝑘𝑡‖𝑓‖^BMO𝐿𝛼≲^{(1−𝑒2−2𝑡)𝑒𝑡2/𝑡𝑥−𝑥/𝑡−∞}𝑒^{(−1/16)𝑧2}𝑑𝑧1+log20𝜌^𝐿𝛼𝑥_𝑘𝑡‖𝑓‖^BMO𝐿𝛼+‖𝑓‖_BMO𝐿𝛼≲𝑒^{−(1/20)((1−𝑒2−2𝑡)𝑒𝑡2/𝑡𝑥−}𝑥𝑡⁾1+log20𝜌^𝐿𝛼𝑥_𝑘𝑡‖𝑓‖^BMO𝐿𝛼+‖𝑓‖_BMO𝐿𝛼≲𝑡²𝜌_𝐿𝛼𝑥_𝑘²1+log20𝜌^𝐿𝛼𝑥_𝑘𝑡‖𝑓‖^BMO𝐿𝛼+‖𝑓‖_BMO𝐿𝛼≲‖𝑓‖_BMO𝐿𝛼.(4.31) The third and fourth inequalities come from the fact (1−𝑒^−2𝑡2)𝑒^𝑡2/𝑥𝑡≲𝑥/𝑡 and 𝑥∼𝑥_𝑘, when 𝑡≤20𝜌_𝐿𝛼(𝑥_𝑘) and 𝑥∈𝐵^∗𝑘. It is notable that, for 𝑡≤20𝜌_𝐿𝛼(𝑥_𝑘) and 𝑥∈𝐵^∗𝑘, the inequality above also implies ||||_{𝑋𝑡2(𝑥)}𝑃_𝑡(𝑥,𝑦)𝑑𝑦||||≤𝐶𝑡𝜌_𝐿𝛼𝑥_𝑘.(4.32) On the other hand, by (a) of Proposition 2.10 and the simple fact that |𝑓₂^𝑗_{𝐵𝑡(𝑥)}−𝑓_{𝐵𝑡(𝑥)}|≤𝐶𝑗‖𝑓‖_BMO𝐿𝛼, we obtain ||||_{𝑋𝑡2(𝑥)}||𝑓(𝑦)−𝑓_{𝐵𝑡(𝑥)}||||𝑃_𝑡(𝑥,𝑦)||𝑑𝑦||||≲^∞01𝑡𝑒^{−|𝑦−𝑥|}