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|2/10𝑡2}≲||𝑓(𝑦)−𝑓_{𝐵𝑡(𝑥)}||𝑑𝑦_∞_𝑗=02^{−𝑗(𝑁−1)}12^𝑗𝑡_{{𝑦>0,|𝑦−𝑥|≤2}^𝑗_𝑡}+||𝑓(𝑦)−𝑓_{2𝑗𝐵𝑡(𝑥)}||𝑑𝑦_∞_𝑗=0≲_∞2^{−𝑗(𝑁−1)}||𝑓_{2𝑗𝐵𝑡(𝑥)}−𝑓_{𝐵𝑡(𝑥)}||_𝑗=0𝑗2^{−𝑗(𝑁−1)}‖𝑓‖_BMO𝐿𝛼≲‖𝑓‖_BMO𝐿𝛼.(4.33) From (4.24), we deduce that |||||_{20𝜌𝐿𝛼(𝑥𝑘)𝑟}||𝑄_𝑡𝑓(𝑥)||²𝑑𝑡𝑡^1/2−^{20𝜌𝐿𝛼(𝑥𝑘)𝑟}||𝑄_𝑡𝑓𝑥₀||²𝑑𝑡𝑡^1/2≲_{20𝜌𝐿𝛼(𝑥𝑘)𝑟}|||||||𝑄_𝑡𝑓(𝑥)−𝑄_𝑡𝑓𝑥₀||²𝑑𝑡𝑡^1/2≲‖𝑓‖^1/2_BMO𝐿𝛼^{20𝜌𝐿𝛼(𝑥𝑘)𝑟}||𝑄_𝑡𝑓(𝑥)−𝑄_𝑡𝑓𝑥₀||𝑑𝑡𝑡^1/2.(4.34) By (4.28) and (4.29), for 𝑧∈𝐵^∗𝑘, we have ^{20𝜌𝐿𝛼(𝑥𝑘)𝑟}_{𝑋𝑡1(𝑧)}||𝑓(𝑦)||||𝑄_𝑡(𝑧,𝑦)||𝑑𝑦𝑑𝑡𝑡≤𝐶‖𝑓‖^BMO𝐿𝛼_{20𝜌𝐿𝛼(𝑥𝑘)𝑟},_{𝑋𝑡2(𝑧)}||𝑓(𝑦)||||𝑄_𝑡(𝑧,𝑦)−𝑃_𝑡(𝑧,𝑦)||𝑑𝑦𝑑𝑡𝑡≤𝐶‖𝑓‖^BMO𝐿𝛼.(4.35) To finish the proof of (4.23), it suffices to show ^{20𝜌𝐿𝛼(𝑥𝑘)𝑟}||||_{𝑋𝑡2(𝑥)}𝑃_𝑡(𝑥,𝑦)−𝑃_𝑡𝑥₀,𝑦𝑓(𝑦)𝑑𝑦||||𝑑𝑡𝑡≤𝐶‖𝑓‖^BMO𝐿𝛼,(4.36)_{20𝜌𝐿𝛼(𝑥𝑘)𝑟}||||_{𝑋𝑡2(𝑥)}𝑃_𝑡𝑥₀,𝑦𝑓(𝑦)𝑑𝑦−_{𝑋𝑡2(𝑥0)}𝑃_𝑡𝑥₀,𝑦𝑓(𝑦)𝑑𝑦||||𝑑𝑡𝑡≤𝐶‖𝑓‖^BMO𝐿𝛼.(4.37) It is easy to check (4.37). Indeed, without loss of generality, we assume 𝑥<𝑥₀. According to (a) of Proposition 2.10, ||||_{𝑋𝑡2(𝑥)}≲𝑃_𝑡𝑥₀,𝑦𝑓(𝑦)𝑑𝑦−_{𝑋𝑡2(𝑥0)}𝑃_𝑡𝑥₀,𝑦𝑓(𝑦)𝑑𝑦_{(1−𝑒2−2𝑡)/𝑒𝑡2𝑥1−𝑒2−2𝑡/𝑒𝑡2𝑥0}||||1𝑡||𝑓(𝑦)||𝑑𝑦𝑥𝑡𝑟²⁰‖𝑓‖_BMO𝐿𝛼.(4.38) Inserting this into (4.37) gives the desired result. Moreover, from the above we also have proved ||||_{𝑋𝑡2(𝑥)}𝑃_𝑡𝑥₀,𝑦𝑓(𝑦)𝑑𝑦−_{𝑋𝑡2(𝑥0)}𝑃_𝑡𝑥₀,𝑦𝑑𝑦||||≤𝐶𝑡𝜌_𝐿𝛼𝑥₀.(4.39)
For 𝑟≤𝑡≤20𝜌_𝐿𝛼(𝑥_𝑘), we write ||||_{𝑋𝑡2(𝑥)}≤||||𝑃_𝑡(𝑥,𝑦)−𝑃_𝑡𝑥₀,𝑦𝑓(𝑦)𝑑𝑦_{𝑋𝑡2(𝑥)}||||=𝑆_𝑡1𝑃_𝑡(𝑥,𝑦)−𝑃_𝑡𝑥₀,𝑦𝑓(𝑦)−𝑓_𝐵𝑑𝑦||||+||𝑓_𝐵||||||_{𝑋𝑡2(𝑥)}𝑃_𝑡(𝑥,𝑦)−𝑃_𝑡𝑥₀,𝑦𝑑𝑦||||(𝑥)+𝑆^𝑡2(𝑥).(4.40) By (b) of Proposition 2.10, we obtain 𝑆^𝑡1(𝑥)≲_{{𝑦>0,|𝑦−𝑥0|>𝑟}}𝑟𝑡1𝑡𝑒^{−𝑐|𝑦−𝑥0|2/𝑡2}||𝑓(𝑦)−𝑓_𝐵||𝑑𝑦+1𝑡_{{𝑦>0,|𝑦−𝑥0|≤𝑟}}≲||𝑓(𝑦)−𝑓_𝐵||𝑑𝑦_∞_𝑗=0_2𝑗+1𝐵𝑟𝑡^1/212^𝑗𝑟^3/2||𝑓(𝑦)−𝑓_2𝑗+1𝐵+𝑓₂^𝑗+1_𝐵−𝑓_𝐵||𝑑𝑦+𝑟𝑡‖𝑓‖^BMO𝐿𝛼≲𝑟𝑡^∞1/2_𝑗=02^−𝑗/2𝑗‖𝑓‖_BMO𝐿𝛼+𝑟𝑡‖𝑓‖^BMO𝐿𝛼≲𝑟𝑡+𝑟𝑡^1/2‖𝑓‖^BMO𝐿𝛼.(4.41)
From the inequality above, we deduce that ^{20𝜌𝐿𝛼(𝑥𝑘)𝑟}𝑆^𝑡1(𝑥)𝑑𝑡𝑡≤^{20𝜌𝐿𝛼(𝑥𝑘)𝑟}𝑟𝑡+𝑟𝑡^1/2𝑑𝑡𝑡‖𝑓‖^BMO𝐿𝛼≤𝐶‖𝑓‖_BMO𝐿𝛼.(4.42)
On the other hand, by using (b) of Proposition 2.10 again, we also have ||||_{𝑋𝑡2(𝑥)}𝑃_𝑡(𝑥,𝑦)−𝑃_𝑡𝑥₀,𝑦𝑑𝑦||||≤𝐶𝑟𝑡.(4.43) This together with (4.32), (4.39), and |𝑓_𝐵|≲(1+log(𝜌_𝐿𝛼(𝑥₀)/𝑟))‖𝑓‖_BMO𝐿𝛼 gives that ^{20𝜌𝐿𝛼(𝑥𝑘)𝑟}𝑆^𝑡2(𝑥)𝑑𝑡𝑡≲^{20𝜌𝐿𝛼(𝑥𝑘)𝑟}𝑟𝑡^2/3𝜌_𝐿𝛼𝑡𝑥_𝑘^1/31+log𝜌^𝐿𝛼𝑥₀𝑟𝑑𝑡𝑡‖𝑓‖^BMO𝐿𝛼≲^{20𝜌𝐿𝛼(𝑥𝑘)𝑟}𝑟𝑡^1/3𝑑𝑡𝑡‖𝑓‖^BMO𝐿𝛼≲‖𝑓‖_BMO𝐿𝛼.(4.44) Thus we finish the proof of (4.36) and, hence, finish the proof of Theorem 4.1.

Acknowledgments

The authors would like to express their gratitude to the referee for many valuable suggestions and revisions which improve the presentation of this paper. Supported by National Natural Science Foundation of China under Grant no. 10990012.