Necessary and Sufficient Conditions for Boundedness of Commutators of the General Fractional Integral Operators on Weighted Morrey Spaces

Abstract

We prove that 𝑏 is in Lip_𝛽(𝜔) if and only if the commutator [𝑏,𝐿^−𝛼/2] of the multiplication operator by 𝑏 and the general fractional integral operator 𝐿^−𝛼/2 is bounded from the weighted Morrey space 𝐿^𝑝,𝑘(𝜔) to 𝐿^{𝑞,𝑘𝑞/𝑝}(𝜔^{1−(1−𝛼/𝑛)𝑞},𝜔), where 0<𝛽<1, 0<𝛼+𝛽<𝑛,1<𝑝<𝑛/(𝛼+𝛽), 1/𝑞=1/𝑝−(𝛼+𝛽)/𝑛, 0≤𝑘<𝑝/𝑞, 𝜔^𝑞/𝑝∈𝐴_1, and 𝑟_𝜔>(1−𝑘)/(𝑝/(𝑞−𝑘)), and here 𝑟_𝜔 denotes the critical index of 𝜔 for the reverse Hölder condition.

1. Introduction and Main Results

Suppose that 𝐿 is a linear operator on 𝐿²(ℝ^𝑛) which generates an analytic semigroup 𝑒^−𝑡𝐿 with a kernel 𝑝_𝑡(𝑥,𝑦) satisfying a Gaussian upper bound, that is, ||𝑝_𝑡(𝑥,𝑦)||≤𝐶𝑡^𝑛/2𝑒^{−𝑐(|𝑥−𝑦|2/𝑡)}(1.1) for 𝑥,𝑦∈ℝ^𝑛 and all 𝑡>0. Since we assume only upper bound on heat kernel 𝑝_𝑡(𝑥,𝑦) and no regularity on its space variables, this property (1.1) is satisfied by a class of differential operator, see [1] for details.

For 0<𝛼<𝑛, the general fractional integral 𝐿^−𝛼/2 of the operator 𝐿 is defined by 𝐿^−𝛼/2𝑓(𝑥)=1Γ(𝛼/2)^∞0𝑒^−𝑡𝐿𝑓𝑑𝑡𝑡^−𝛼/2+1(𝑥).(1.2)

Note that, if 𝐿=−Δ is the Laplacian on ℝ^𝑛, then, 𝐿^−𝛼/2 is the classical fractional integral 𝐼_𝛼 which plays important roles in many fields. Let 𝑏 be a locally integrable function on ℝ^𝑛, the commutator of 𝑏 and 𝐿^−𝛼/2 is defined by 𝑏,𝐿^−𝛼/2𝑓(𝑥)=𝑏(𝑥)𝐿^−𝛼/2𝑓(𝑥)−𝐿^−𝛼/2(𝑏𝑓)(𝑥).(1.3)

For the special case of 𝐿=−Δ, many results have been produced. Paluszyński [2] obtained that 𝑏∈Lip_𝛽(ℝ^𝑛) if the commutator [𝑏,𝐼_𝛼] is bounded from 𝐿^𝑝(ℝ^𝑛) to 𝐿^𝑟(ℝ^𝑛), where 1<𝑝<𝑟<∞,0<𝛽<1 and 1/𝑝−1/𝑟=(𝛼+𝛽)/𝑛 with 𝑝<𝑛/(𝛼+𝛽). Shirai [3] proved that 𝑏∈Lip_𝛽(ℝ^𝑛) if and only if the commutator [𝑏,𝐼_𝛼] is bounded from the classical Morrey spaces 𝐿^𝑝,𝜆(ℝ^𝑛) to 𝐿^𝑞,𝜆(ℝ^𝑛) for 1<𝑝<𝑞<∞,0<𝛼,0<𝛽<1, and 0<𝛼+𝛽=(1/𝑝−1/𝑞)(𝑛−𝜆)<𝑛 or 𝐿^𝑝,𝜆(ℝ^𝑛) to 𝐿^𝑞,𝜇(ℝ^𝑛) for 1<𝑝<𝑞<∞,0<𝛼,0<𝛽<1,0<𝛼+𝛽=(1/𝑝−1/𝑞)<𝑛,0<𝜆<𝑛−(𝛼+𝛽)𝑝, and 𝜇/𝑞=𝜆/𝑝. Wang [4] established some weighted boundedness of properties of commutator [𝑏,𝐼_𝛼] on the weighted Morrey spaces 𝐿^𝑝,𝑘 under appropriated conditions on the weight 𝜔, where the symbol 𝑏 belongs to (weighted) Lipschitz spaces. The weighted Morrey space was first introduced by Komori and Shirai [5]. For the general case, Wang [6] proved that if 𝑏∈Lip_𝛽(ℝ^𝑛), then the commutator [𝑏,𝐿^−𝛼/2] is bounded from 𝐿^𝑝(𝜔^𝑝) to 𝐿^𝑞(𝜔^𝑞), where 0<𝛽<1,0<𝛼+𝛽<𝑛,1<𝑝<𝑛/(𝛼+𝛽),1/𝑝−1/𝑞=(𝛼+𝛽)/𝑛, and 𝜔^𝑞∈𝐴₁.

The purpose of this paper is to give necessary and sufficient conditions for boundedness of commutators of the general fractional integrals with 𝑏∈Lip_𝛽(𝜔) (the weighted Lipschitz space). Our theorems are the following.

Theorem 1.1. Let 0<𝛽<1, 0<𝛼+𝛽<𝑛,1<𝑝<𝑛/(𝛼+𝛽), 1/𝑞=1/𝑝−(𝛼+𝛽)/𝑛,0≤𝑘<min{𝑝/𝑞,𝑝𝛽/𝑛}, and 𝜔^𝑞∈𝐴₁. Then one has the following. (a)If 𝑏∈Lip_𝛽(ℝ^𝑛), then [𝑏,𝐿^−𝛼/2] is bounded from 𝐿^𝑝,𝑘(𝜔^𝑝,𝜔^𝑞) to 𝐿^{𝑞,𝑘𝑞/𝑝}(𝜔^𝑞); (b)If [𝑏,𝐿^−𝛼/2] is bounded from 𝐿^𝑝,𝑘(𝜔^𝑝,𝜔^𝑞) to 𝐿^{𝑞,𝑘𝑞/𝑝}(𝜔^𝑞), then 𝑏∈Lip_𝛽(ℝ^𝑛).

Theorem 1.2. Let 0<𝛽<1, 0<𝛼+𝛽<𝑛,1<𝑝<𝑛/(𝛼+𝛽), 1/𝑞=1/𝑝−(𝛼+𝛽)/𝑛,0≤𝑘<𝑝/𝑞,𝜔^𝑞/𝑝∈𝐴₁, and 𝑟_𝜔>(1−𝑘)/(𝑝/(𝑞−𝑘)), where 𝑟_𝜔 denotes the critical index of 𝜔 for the reverse Hölder condition. Then one has the following. (a)If 𝑏∈Lip_𝛽(𝜔), then [𝑏,𝐿^−𝛼/2] is bounded from 𝐿^𝑝,𝑘(𝜔) to 𝐿^{𝑞,𝑘𝑞/𝑝}(𝜔^{1−(1−𝛼/𝑛)𝑞},𝜔); (b)If [𝑏,𝐿^−𝛼/2] is bounded from 𝐿^𝑝,𝑘(𝜔) to 𝐿^{𝑞,𝑘𝑞/𝑝}(𝜔^{1−(1−𝛼/𝑛)𝑞},𝜔), then 𝑏∈Lip_𝛽(𝜔).

Our results not only extend the results of [4] from (−Δ) to a general operator 𝐿, but also characterize the (weighted) Lipschitz spaces by means of the boundedness of [𝑏,𝐿^−𝛼/2] on the weighted Morrey spaces, which extend the results of [4, 6]. The basic tool is based on a modification of sharp maximal function 𝑀^#𝐿 introduced by [7].

Throughout this paper all notation is standard or will be defined as needed. Denote the Lebesgue measure of 𝐵 by |𝐵| and the weighted measure of 𝐵 by 𝜔(𝐵), where 𝜔(𝐵)=∫_𝐵𝜔(𝑥)𝑑𝑥. For a measurable set 𝐸, denote by 𝜒_𝐸 the characteristic function of 𝐸. For a real number 𝑝, 1<𝑝<∞, let 𝑝′ be the dual of 𝑝 such that 1/𝑝+1/𝑝′=1. The letter 𝐶 will be used for various constants, and may change from one occurrence to another.

2. Some Preliminaries

A nonnegative function 𝜔 defined on ℝ^𝑛 is called weight if it is locally integral. A weight 𝜔 is said to belong to the Muckenhoupt class 𝐴_𝑝(ℝ^𝑛) for 1<𝑝<∞, if there exists a positive constant 𝐶 such that ||𝐵1||_𝐵𝜔(𝑥)𝑑𝑥1||𝐵||_𝐵𝜔(𝑥)^{−1/(𝑝−1)}𝑑𝑥^𝑝−1≤𝐶,(2.1) for every ball 𝐵⊂ℝ^𝑛. The class 𝐴₁(ℝ^𝑛) is defined replacing the above inequality by ||𝐵1||_𝐵𝜔(𝑥)𝑑𝑥≤𝐶essinf_𝑥∈𝐵𝜔(𝑥).(2.2) When 𝑝=∞,𝜔∈𝐴_∞, if there exist positive constants 𝛿 and 𝐶 such that given a ball 𝐵 and 𝐸 is a measurable subset of 𝐵, then 𝜔(𝐸)𝜔(𝐵)≤𝐶||||𝐸𝐵||||^𝛿.(2.3)

A weight function 𝜔 belongs to 𝐴_𝑝,𝑞 for 1<𝑝<𝑞<∞ if for every ball 𝐵 in ℝ^𝑛, there exists a positive constant 𝐶 which is independent of 𝐵 such that |𝐵1|_𝐵𝜔(𝑥)^𝑞𝑑𝑥^1/𝑞|𝐵1|_𝐵𝜔(𝑥)^−𝑝′𝑑𝑥^1/𝑝′≤𝐶.(2.4)

From the definition of 𝐴_𝑝,𝑞, we can get that 𝜔∈𝐴_𝑝,𝑞,iff𝜔^𝑞∈𝐴_{1+𝑞/𝑝′}.(2.5) Since 𝜔^𝑞/𝑝∈𝐴₁, then by (2.5), we have 𝜔^1/𝑝∈𝐴_𝑝,𝑞.

A weight function 𝜔 belongs to the reverse Hölder class RH_𝑟 if there exist two constants 𝑟>1 and 𝐶>0 such that the following reverse Hölder inequality, |𝐵1|_𝐵𝜔(𝑥)^𝑟𝑑𝑥^1/𝑟≤𝐶1||𝐵||_𝐵𝜔(𝑥)𝑑𝑥,(2.6) holds for every ball 𝐵 in ℝ^𝑛.

It is well known that if 𝜔∈𝐴_𝑝 with 1≤𝑝<∞, then there exists 𝑟>1 such that 𝜔∈RH_𝑟. It follows from Hölder inequality that 𝜔∈RH_𝑟 implies 𝜔∈RH_𝑠 for all 1<𝑠<𝑟. Moreover, if 𝜔∈RH_𝑟,𝑟>1, then we have 𝜔∈RH_𝑟+𝜀 for some 𝜖>0. We thus write 𝑟_𝑤=sup{𝑟>1∶𝜔∈RH_𝑟} to denote the critical index of 𝜔 for the reverse Hölder condition. For more details on Muckenhoupt class 𝐴_𝑝,𝑞, we refer the reader to [8–10].

Definition 2.1 (see [5]). Let 1≤𝑝<∞ and 0≤𝑘<1. Then for two weights 𝜇 and 𝜈, the weighted Morrey space is defined by 𝐿^𝑝,𝑘(𝜇,𝜈)=𝑓∈𝐿^𝑝loc(𝜇)∶‖𝑓‖_𝐿^𝑝,𝑘_(𝜇,𝜈)<∞,(2.7) where ‖𝑓‖_𝐿^𝑝,𝑘_(𝜇,𝜈)=sup_𝐵𝜈(𝐵1)^𝑘_𝐵||𝑓(𝑥)||^𝑝𝜇(𝑥)𝑑𝑥^1/𝑝,(2.8) and the supremum is taken over all balls 𝐵 in ℝ^𝑛.

If 𝜇=𝜈, then we have the classical Morrey space 𝐿^𝑝,𝑘(𝜇) with measure 𝜇. When 𝑘=0, then 𝐿^𝑝,0(𝜇)=𝐿^𝑝(𝜇) is the Lebesgue space with measure 𝜇.

Definition 2.2 (see [11]). Let 1≤𝑝<∞, 0<𝛽<1, and 𝜔∈𝐴_∞. A locally integral function 𝑏 is said to be in Lip^𝑝𝛽(𝜔) if ‖𝑏‖_Lip^𝑝𝛽_(𝜔)=sup_𝐵1𝜔(𝐵)^𝛽/𝑛𝜔(1𝐵)_𝐵||𝑏(𝑥)−𝑏_𝐵||^𝑝𝜔(𝑥)^1−𝑝𝑑𝑥^1/𝑝≤𝐶<∞,(2.9) where 𝑏_𝐵=|𝐵|⁻¹∫_𝐵𝑏(𝑦)𝑑𝑦 and the supremum is taken over all ball 𝐵⊂𝑅^𝑛. When 𝑝=1, we denote Lip^𝑝𝛽(𝜔) by Lip_𝛽(𝜔).

Obviously, for the case 𝜔=1, then the Lip^𝑝𝛽(𝜔) space is the classical Lip^𝑝𝛽 space.

Remark 2.3. Let 𝜔∈𝐴₁, García-Cuerva [11] proved that the spaces ‖𝑓‖_Lip^𝑝𝛽_(𝜔) coincide, and the norms of ||⋅||_Lip^𝑝𝛽_(𝜔) are equivalent with respect to different values of provided that 1≤𝑝<∞.

Given a locally integrable function 𝑓 and 𝛽, 0≤𝛽<𝑛, define the fractional maximal function by 𝑀_𝛽,𝑟𝑓(𝑥)=sup_𝑥∈𝐵||𝐵||1^{1−𝛽𝑟/𝑛}_𝐵||𝑓(𝑦)||^𝑟𝑑𝑦^1/𝑟,𝑟≥1,(2.10) when 0<𝛽<𝑛. If 𝛽=0 and 𝑟=1, then 𝑀_0,1𝑓=𝑀𝑓 denotes the usual Hardy-Littlewood maximal function.

Let 𝜔 be a weight. The weighted maximal operator 𝑀_𝜔 is defined by 𝑀_𝜔𝑓(𝑥)=sup_𝑥∈𝐵1𝜔(𝐵)_𝐵||𝑓(𝑦)||𝑑𝑦.(2.11) The fractional weighted maximal operator 𝑀_{𝛽,𝑟,𝜔} is defined by 𝑀_{𝛽,𝑟,𝜔}𝑓(𝑥)=sup_𝑥∈𝐵𝜔(𝐵)1^{1−𝛽𝑟/𝑛}_𝐵||𝑓(𝑦)||^𝑟𝜔(𝑦)𝑑𝑦^1/𝑟,(2.12) where 0≤𝛽<𝑛 and 𝑟≥1. For any 𝑓∈𝐿^𝑝(ℝ^𝑛),𝑝≥1, the sharp maximal function 𝑀^#𝐿𝑓 associated the generalized approximations to the identity {𝑒^−𝑡𝐿,𝑡>0} is given by Martell [7] as follows: 𝑀^#𝐿𝑓(𝑥)=sup_𝑥∈𝐵1||𝐵||_𝐵||𝑓(𝑦)−𝑒^{−𝑡𝐵𝐿}𝑓(𝑦)||𝑑𝑦,(2.13) where 𝑡_𝐵=𝑟^2𝐵 and 𝑟_𝐵 is the radius of the ball 𝐵. For 0<𝛿<1, we introduce the 𝛿-sharp maximal operator 𝑀^#𝐿,𝛿 as 𝑀^#𝐿,𝛿𝑓=𝑀^#𝐿||𝑓||^𝛿^1/𝛿,(2.14) which is a modification of the sharp maximal operator 𝑀^# of Stein and Murphy [9]. Set 𝑀_𝛿𝑓=𝑀(|𝑓|^𝛿)^1/𝛿. Using the same methods as those of [9, 12], we can get the following.

Lemma 2.4. Assume that the semigroup 𝑒^−𝑡𝐿 has a kernel 𝑝_𝑡(𝑥,𝑦) which satisfies the upper bound (1.1). Let 𝜆>0 and 𝑓∈𝐿^𝑝(ℝ^𝑛) for some 1<𝑝<∞. Suppose that 𝜔∈𝐴_∞, then for every 0<𝜂<1, there exists a real number 𝛾>0 independent of 𝛾,𝑓 such that one has the following weighted version of the local good 𝜆 inequality, for 𝜂>0, 𝐴>1, 𝜔𝑥∈ℝ^𝑛∶𝑀_𝛿𝑓>𝐴𝜆,𝑀^#𝐿,𝛿𝑓(𝑥)≤𝛾𝜆≤𝜂𝜔𝑥∈ℝ^𝑛∶𝑀_𝛿𝑓(𝑥)>𝜆,(2.15) where 𝐴>1 is a fixed constant which depends only on 𝑛.

If 𝜇,𝜈∈𝐴_∞,1<𝑝<∞,0≤𝑘<1, then ‖𝑓‖_𝐿^𝑝,𝑘_(𝜇,𝜈)≤‖‖𝑀_𝛿𝑓‖‖_{𝐿𝑝,𝑘(𝜇,𝜈)}≤𝐶‖‖𝑀^#𝐿,𝛿𝑓‖‖_{𝐿𝑝,𝑘(𝜇,𝜈)}.(2.16) In particular, when 𝜇=𝜈=𝜔 and 𝜔∈𝐴_∞, we have ‖𝑓‖_𝐿^𝑝,𝑘_(𝜔)≤‖‖𝑀_𝛿𝑓‖‖_{𝐿𝑝,𝑘(𝜔)}≤𝐶‖‖𝑀^#𝐿,𝛿𝑓‖‖_{𝐿𝑝,𝑘(𝜔)}.(2.17)

3. Proof of Theorem 1.1

To prove Theorem 1.1, we need the following lemmas.

Lemma 3.1 (see [1]). Assume that the semigroup 𝑒^−𝑡𝐿 has a kernel 𝑝_𝑡(𝑥,𝑦) which satisfies the upper bound (1.1). Then for 0<𝛼<1, the difference operator 𝐿^−𝛼/2−𝑒^−𝑡𝐿𝐿^−𝛼/2 has an associated kernel 𝐾_𝛼,𝑡(𝑥,𝑦) which satisfies 𝐾_𝛼,𝑡(𝑥,𝑦)≤𝐶||𝑥−𝑦||^𝑛−𝛼𝑡||𝑥−𝑦||².(3.1)

Lemma 3.2 (see [4]). Let 0<𝛼+𝛽<𝑛, 1<𝑝<𝑛/(𝛼+𝛽),1/𝑞=1/𝑝−(𝛼+𝛽)/𝑛, and 𝜔∈𝐴₁. Then for every 0<𝑘<𝑝/𝑞 and 1<𝑟<𝑝, one has ‖‖𝑀_{𝛼+𝛽,𝑟}𝑓‖‖_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞)}≤𝐶‖𝑓‖_𝐿^𝑝,𝑞_{(𝜔𝑝,𝜔𝑞)}.(3.2)

Lemma 3.3 (see [5]). Let 0<𝛽<𝑛, 1<𝑝<𝑛/𝛽,1/𝑠=1/𝑝−𝛽/𝑛, and 𝜔∈𝐴_𝑝,𝑠. Then for every 0<𝑘<𝑝/𝑠, one has ‖‖𝑀_𝛽,1𝑓‖‖_{𝐿𝑠,𝑘𝑠/𝑝(𝜔𝑠)}≤𝐶‖𝑓‖_𝐿^𝑝,𝑘_{(𝜔𝑝,𝜔𝑠)}.(3.3)

Lemma 3.4 (see [4]). Let 0<𝛼+𝛽<𝑛, 1<𝑝<𝑛/(𝛼+𝛽),1/𝑞=1/𝑝−𝛼/𝑛, 1/𝑠=1/𝑞−𝛽/𝑛, and 𝜔^𝑞∈𝐴₁. Then for every 0<𝑘<𝑝/𝑠, one has ‖‖𝑀_𝛽,1𝑓‖‖_{𝐿𝑠,𝑘𝑠/𝑝(𝜔𝑠)}≤𝐶‖𝑓‖_𝐿^{𝑞,𝑘𝑞/𝑝}_{(𝜔𝑞,𝜔𝑠)}.(3.4)

Lemma 3.5. Let 0<𝛼+𝛽<𝑛, 1<𝑝<𝑛/(𝛼+𝛽),1/𝑞=1/𝑝−𝛼/𝑛, 1/𝑠=1/𝑞−𝛽/𝑛, and 𝜔^𝑞∈𝐴₁. Then for every 0<𝑘<𝑝𝛽/𝑛, one has ‖‖𝐿^−𝛼/2𝑓‖‖_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞,𝜔𝑠)}≤𝐶‖𝑓‖_𝐿^𝑝,𝑘_{(𝜔𝑝,𝜔𝑠)}.(3.5)

Proof. Since the semigroup 𝑒^−𝑡𝐿 has a kernel 𝑝_𝑡(𝑥,𝑦) which satisfies the upper bound (1.1), it is easy to check that 𝐿^−𝛼/2𝑓(𝑥)≤𝐶𝐼_𝛼(|𝑓|)(𝑥) for all 𝑥∈ℝ^𝑛. Together with the result (cf. [4]), that is, ‖‖𝐼_𝛼𝑓‖_𝐿^{𝑞,𝑘𝑞/𝑝}_{(𝜔𝑞,𝜔𝑠)}≤𝐶‖‖𝑓‖_𝐿^𝑝,𝑘_{(𝜔𝑝,𝜔𝑠)},(3.6) we can get the desired result.

Remark 3.6. Using the boundedness property of 𝐼_𝛼, we also know 𝐿^−𝛼/2 is bounded from 𝐿¹ to weak 𝐿^{𝑛/(𝑛−𝛼)}. It is easy to check that Lemmas 3.2–3.5 also hold when 𝑘=0.

The following lemma plays an important role in the proof of Theorem 1.1.

Lemma 3.7. Let 0<𝛿<1,0<𝛼<𝑛,0<𝛽<1, and 𝑏∈Lip_𝛽(ℝ^𝑛). Then for all 𝑟>1 and for all 𝑥∈ℝ^𝑛, one has 𝑀^#𝐿,𝛿≤𝐶𝑏,‖𝐿𝑏‖^−𝛼/2𝑓_{Lip𝛽(ℝ𝑛)}(𝑥)𝑀_𝛽,1𝐿^−𝛼/2𝑓(𝑥)+𝑀_{𝛼+𝛽,𝑟}𝑓(𝑥)+𝑀_𝛼+𝛽,1𝑓(𝑥).(3.7)

The same method of proof as that of Lemma 4.7 (see below), one omits the details.

Proof of Theorem 1.1. We first prove (a). We only prove Theorem 1.1 in the case 0<𝛼<1. For the general case 0<𝛼<𝑛, the method is the same as that of [1]. We omit the details.
For 0<𝛼+𝛽<𝑛 and 1<𝑝<𝑛/(𝛼+𝛽), we can find a number 𝑟 such that 1<𝑟<𝑝. By (2.17) and Lemma 3.7, we obtain the following: ‖‖𝑏,𝐿^−𝛼/2𝑓‖‖_𝐿^{𝑞,𝑘𝑞/𝑝}_(𝜔𝑞)≤𝐶‖‖𝑀^#𝐿,𝛿𝑏,𝐿^−𝛼/2𝑓‖‖_𝐿^{𝑞,𝑘𝑞/𝑝}_(𝜔𝑞)≤𝐶‖𝑏‖_Lip𝛽(ℝ^𝑛₎‖‖𝑀^𝛽,1𝐿^−𝛼/2𝑓‖‖_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞)}+‖‖𝑀_{𝛼+𝛽,𝑟}𝑓‖‖_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞)}+‖‖𝑀_𝛼+𝛽,1𝑓‖‖_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞)}.(3.8) Let 1/𝑞₁=1/𝑝−𝛼/𝑛 and 1/𝑞=1/𝑞₁−𝛽/𝑛. Since 𝜔^𝑞∈𝐴₁, then by (2.5), we have 𝜔∈𝐴_𝑝,𝑞. Since 0<𝑘<min{𝑝/𝑞,𝑝𝛽/𝑛}, by Lemmas 3.2–3.5, we yield that ‖‖𝑏,𝐿^−𝛼/2𝑓‖‖_𝐿^{𝑞,𝑘𝑞/𝑝}_(𝜔𝑞)≤𝐶‖𝑏‖_Lip𝛽(ℝ^𝑛₎‖‖𝐿^−𝛼/2𝑓‖‖_{𝐿𝑞11,𝑘𝑞/𝑝(𝜔𝑞1,𝜔𝑞)}≤𝐶‖𝑏‖_{Lip𝛽(ℝ𝑛)}+‖𝑓‖_𝐿^𝑝,𝑘_{(𝜔𝑝,𝜔𝑞)}‖𝑓‖_𝐿^𝑝,𝑘_{(𝜔𝑝,𝜔𝑞)}.(3.9)
Now we prove (𝑏). Let 𝐿=−Δ be the Laplacian on ℝ^𝑛, then 𝐿^−𝛼/2 is the classical fractional integral 𝐼_𝛼. Let 𝑘=0 and weight 𝜔≡1, then 𝐿^𝑝,𝑘(𝜔^𝑝,𝜔^𝑞)=𝐿^𝑝 and 𝐿^{𝑞,𝑘𝑞/𝑝}(𝜔^𝑞,𝜔)=𝐿^𝑞. From [2], the (𝐿^𝑝,𝐿^𝑞) boundedness of [𝑏,𝐼_𝛼] implies that 𝑏∈Lip_𝛽(ℝ^𝑛).
Thus Theorem 1.1 is proved.

4. Proof of Theorem 1.2

We also need some Lemmas to prove Theorem 1.2.

Lemma 4.1 (see [4]). Let 0<𝛼+𝛽<𝑛,1<𝑝<𝑛/(𝛼+𝛽),1/𝑞=1/𝑝−𝛼/𝑛,1/𝑠=1/𝑞−𝛽/𝑛, and 𝜔^𝑠/𝑝∈𝐴₁. Then if 0<𝑘<𝑝/𝑠 and 𝑟_𝜔>1/(𝑝/(𝑞−𝑘)), one has ‖‖𝑀_𝛽,1𝑓‖‖_{𝐿𝑠,𝑘𝑠/𝑝(𝜔𝑠/𝑝,𝜔)}≤𝐶‖𝑓‖_𝐿^{𝑞,𝑘𝑞/𝑝}_{(𝜔𝑞/𝑝,𝜔)}.(4.1)

Lemma 4.2 (see [4]). Let 0<𝛼<𝑛,1<𝑝<𝑛/𝛼,1/𝑞=1/𝑝−𝛼/𝑛, and 𝜔^𝑞/𝑝∈𝐴₁. Then if 0<𝑘<𝑝/𝑞 and 𝑟_𝜔>(1−𝑘)/(𝑝/(𝑞−𝑘)), one has ‖‖𝑀_𝛼,1𝑓‖‖_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞/𝑝,𝜔)}≤𝐶‖𝑓‖_𝐿^𝑝,𝑘_(𝜔).(4.2)

Lemma 4.3 (see [4]). Let 0<𝛼<𝑛,1<𝑝<𝑛/𝛼,1/𝑞=1/𝑝−𝛼/𝑛, 0<𝑘<𝑝/𝑞, and 𝜔∈𝐴_∞. For any 1<𝑟<𝑝, one has ‖‖𝑀_{𝛼,𝑟,𝜔}𝑓‖‖_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞/𝑝,𝜔)}≤𝐶‖𝑓‖_𝐿^𝑝,𝑘_(𝜔).(4.3)

Lemma 4.4. Let 0<𝛼<𝑛,1<𝑝<𝑛/𝛼,1/𝑞=1/𝑝−𝛼/𝑛, and 𝜔^𝑞/𝑝∈𝐴₁. Then if 0<𝑘<𝑝/𝑞 and 𝑟_𝜔>(1−𝑘)/(𝑝/(𝑞−𝑘)), one has ‖‖𝐿^−𝛼/2𝑓‖‖_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞/𝑝,𝜔)}≤𝐶‖𝑓‖_𝐿^𝑝,𝑘_(𝜔).(4.4)

Proof. As before, we know that 𝐿^−𝛼/2𝑓(𝑥)≤𝐶𝐼_𝛼(|𝑓|)(𝑥) for all 𝑥∈ℝ^𝑛. Using the boundedness property of 𝐼_𝛼 on weighted Morrey space (cf. [4]), we have ‖‖𝐿^−𝛼/2𝑓‖‖_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞/𝑝,𝜔)}≤‖‖𝐼_𝛼𝑓‖‖_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞/𝑝,𝜔)}≤𝐶‖𝑓‖_𝐿^𝑝,𝑘_(𝜔),(4.5) where 1<𝑝<𝑛/𝛼 and 1/𝑞=1/𝑝−𝛼/𝑛.

Remark 4.5. It is easy to check that the above lemmas also hold for 𝑘=0.

Lemma 4.6. Assume that the semigroup 𝑒^−𝑡𝐿 has a kernel 𝑝_𝑡(𝑥,𝑦) which satisfies the upper bound (1.1), and let 𝑏∈Lip_𝛽(𝜔),𝜔∈𝐴₁. Then, for every function 𝑓∈𝐿^𝑝(ℝ^𝑛),𝑝>1,𝑥∈ℝ^𝑛, and 1<𝑟<∞, one has sup_𝑥∈𝐵1||𝐵||_𝐵||𝑒^{−𝑡𝐵𝐿}𝑏(𝑦)−𝑏_2𝐵𝑓(𝑦)||𝑑𝑦≤𝐶‖𝑏‖_{Lip𝛽(𝜔)}𝜔(𝑥)𝑀_{𝛽,𝑟,𝜔}𝑓(𝑥).(4.6)

Proof. Fix 𝑓∈𝐿^𝑝(ℝ^𝑛),1<𝑝<∞ and 𝑥∈𝐵. Then, 1||𝐵||_𝐵≤||𝑒1^{−𝑡𝐵𝐿}𝑏−𝑏_2𝐵𝑓(𝑦)||𝑑𝑦||𝐵||_𝐵_ℝ𝑛≤1||𝑝_𝑡𝐵(𝑦,𝑧)||||𝑏(𝑧)−𝑏_2𝐵𝑓(𝑧)||𝑑𝑧𝑑𝑦||𝐵||_𝐵_2𝐵+1||𝑝_𝑡𝐵(𝑦,𝑧)||||𝑏(𝑧)−𝑏_2𝐵𝑓(𝑧)||𝑑𝑧𝑑𝑦||𝐵||_{𝐵∞𝑘=1}_{2𝑘+1𝐵⧵2𝑘𝐵}≐ℳ+𝒩.||𝑝_𝑡𝐵(𝑦,𝑧)||||𝑏(𝑧)−𝑏_2𝐵𝑓(𝑧)||𝑑𝑧𝑑𝑦(4.7) It follows from 𝑦∈𝐵 and 𝑧∈2𝐵 that ||𝑝_𝑡𝐵(𝑦,𝑧)||≤𝐶𝑡^{𝐵−𝑛/2}≤𝐶1||2𝐵||.(4.8) Thus, Hölder's inequality and Definition 2.2 lead to the following: ℳ≤𝐶1||2𝐵||_2𝐵≤𝐶1||𝑏(𝑧)−𝑏_2𝐵𝑓(𝑧)||𝑑𝑧||2𝐵||_2𝐵||𝑏(𝑧)−𝑏_2𝐵||^𝑟𝜔(𝑧)^1−𝑟𝑑𝑧^1/𝑟_2𝐵||𝑓(𝑧)||^𝑟𝜔(𝑧)𝑑𝑧^1/𝑟≤𝐶‖𝑏‖_{Lip𝛽(𝜔)}1||2𝐵||𝜔(2𝐵)^{𝛽/𝑛+1/𝑟′}𝜔(2𝐵)^1/𝑟𝜔(12𝐵)_2𝐵||𝑓(𝑧)||^𝑟𝜔(𝑧)𝑑𝑧^1/𝑟≤𝐶‖𝑏‖_{Lip𝛽(𝜔)}1||2𝐵||𝜔(2𝐵)^𝛽/𝑛+1𝜔(12𝐵)_2𝐵||𝑓(𝑧)||^𝑟𝜔(𝑧)𝑑𝑧^1/𝑟≤𝐶‖𝑏‖_{Lip𝛽(𝜔)}𝜔(𝑥)1𝜔(2𝐵)^{1−𝛽𝑟/𝑛}_2𝐵||𝑓(𝑧)||^𝑟𝜔(𝑧)𝑑𝑧^1/𝑟≤𝐶‖𝑏‖_{Lip𝛽(𝜔)}𝜔(𝑥)𝑀_{𝛽,𝑟,𝜔}𝑓(𝑥).(4.9) Moreover, for any 𝑦∈𝐵 and 𝑧∈2^𝑘+1𝐵⧵2^𝑘𝐵, we have |𝑦−𝑧|≥2^𝑘−1𝑟_𝐵 and |𝑝_𝑡𝐵(𝑦,𝑧)|≤𝐶(𝑒^{−𝑐22(𝑘−1)}2^(𝑘+1)𝑛/|2^𝑘+1𝐵|)𝒩=1||𝐵||_{𝐵∞𝑘=1}_{2𝑘+1𝐵⧵2𝑘𝐵}≤𝐶_∞||𝑝_𝑡𝐵(𝑦,𝑧)||||𝑏(𝑧)−𝑏_2𝐵𝑓(𝑧)||𝑑𝑧𝑑𝑦_𝑘=1𝑒^{−𝑐22(𝑘−1)}2^(𝑘+1)𝑛||2^𝑘+1𝐵||₂^𝑘+1_𝐵≤𝐶_∞||𝑏(𝑧)−𝑏_2𝐵𝑓(𝑧)||𝑑𝑧_𝑘=1𝑒^{−𝑐22(𝑘−1)}2^(𝑘+1)𝑛||2^𝑘+1𝐵||₂^𝑘+1_𝐵+𝐶_∞||𝑏(𝑧)−𝑏_2𝑘+1𝐵𝑓(𝑧)||𝑑𝑧_𝑘=1𝑒^{−𝑐22(𝑘−1)}2^(𝑘+1)𝑛||2^𝑘+1𝐵||₂^𝑘+1_𝐵≐𝒩₁||𝑏_2𝑘+1𝐵−𝑏_2𝐵𝑓(𝑧)||𝑑𝑧+𝒩₂.(4.10) We will estimate the values of terms 𝒩₁ and 𝒩₂, respectively.
Using Hölder's inequality and Remark 2.3, we have the following: 𝒩₁≤𝐶^∞_𝑘=1𝑒^{−𝑐22(𝑘−1)}2^(𝑘+1)𝑛×_2𝑘+1𝐵||2^𝑘+1𝐵||||𝑏(𝑧)−𝑏_2𝑘+1𝐵||^𝑟𝜔(𝑧)^1−𝑟𝑑𝑧^1/𝑟_2𝑘+1𝐵||𝑓(𝑧)||^𝑟𝜔(𝑧)𝑑𝑧^1/𝑟≤𝐶^∞_𝑘=12^(𝑘+1)𝑛𝑒^{−𝑐22(𝑘−1)}×‖𝑏‖_{Lip𝛽(𝜔)}𝜔2^𝑘+1𝐵||2^𝑘+1𝐵||𝜔2^𝑘+11𝐵^{1−𝛽𝑟/𝑛}_2𝑘+1𝐵||𝑓(𝑧)||^𝑟𝜔(𝑧)𝑑𝑧^1/𝑟≤𝐶‖𝑏‖_{Lip𝛽(𝜔)}𝜔(𝑥)𝑀_{𝛽,𝑟,𝜔}𝑓(𝑥).(4.11)
By a simple calculation, we have ||𝑏_2𝑘+1𝐵−𝑏_2𝐵||≤𝐶𝑘𝜔(𝑥)‖𝑏‖_{Lip𝛽(𝜔)}𝜔2^𝑘+1𝐵^𝛽/𝑛.(4.12) Since 𝜔∈𝐴₁, by the Hölder inequality, we get 𝒩₂≤𝐶^∞_𝑘=12^(𝑘+1)𝑛𝑒^{−𝑐22(𝑘−1)}𝑘𝜔2^𝑘+1𝐵^𝛽/𝑛||2^𝑘+1𝐵||𝜔(𝑥)‖𝑏‖^{Lip𝛽(𝜔)}_2𝑘+1𝐵≤𝐶_∞||𝑓(𝑧)||𝑑𝑧_𝑘=1𝑘2^(𝑘+1)𝑛𝑒^{−𝑐22(𝑘−1)}𝜔(𝑥)‖𝑏‖_{Lip𝛽(𝜔)}𝜔2^𝑘+1𝐵^𝛽/𝑛||2^𝑘+1𝐵||^1/𝑟_2𝑘+1𝐵||𝑓(𝑧)||^𝑟𝑑𝑧^1/𝑟=𝐶^∞_𝑘=1𝑘2^(𝑘+1)𝑛𝑒^{−𝑐22(𝑘−1)}×𝜔(𝑥)‖𝑏‖_{Lip𝛽(𝜔)}𝜔(2^𝑘+1𝐵)|2^𝑘+1𝐵|𝜔2^𝑘+11𝐵^{1−𝛽𝑟/𝑛}_2𝑘+1𝐵||𝑓(𝑧)||^𝑟𝑑𝑧^1/𝑟≤𝐶^∞_𝑘=1𝑘2^(𝑘+1)𝑛𝑒^{−𝑐22(𝑘−1)}𝜔(𝑥)‖𝑏‖_{Lip𝛽(𝜔)}𝜔(2^𝑘+11𝐵)^{1−𝛽𝑟/𝑛}_2𝑘+1𝐵||𝑓(𝑧)||^𝑟𝜔(𝑧)𝑑𝑧^1/𝑟≤𝐶‖𝑏‖_{Lip𝛽(𝜔)}𝜔(𝑥)𝑀_{𝛽,𝑟,𝜔}𝑓(𝑥).(4.13)
Thus, Lemma 4.6 is proved.

Lemma 4.7. Let 0<𝛿<1, 0<𝛼<1,𝜔∈𝐴₁, and 𝑏∈Lip_𝛽(𝜔). Then for all 𝑟>1 and for all 𝑥∈ℝ^𝑛, one has 𝑀^#𝐿,𝛿≤𝐶𝑏,‖𝐿𝑏‖^−𝛼/2𝑓_{Lip𝛽(𝜔)}(𝑥)+𝜔(𝑥)_{1−𝛼/𝑛}×𝜔(𝑥)^1+𝛽/𝑛𝑀_𝛽,1𝐿^−𝛼/2𝑓(𝑥)𝑀_{𝛼+𝛽,𝑟,𝜔}𝑓(𝑥)+𝜔(𝑥)^1+𝛽/𝑛𝑀_𝛼+𝛽,1𝑓(𝑥).(4.14)

Proof. For any given 𝑥∈ℝ^𝑛, fix a ball 𝐵=𝐵(𝑥₀,𝑟_𝐵) which contains 𝑥. We decompose 𝑓=𝑓₁+𝑓₂, where 𝑓₁=𝑓𝜒_2𝐵. Observe that 𝑒_{−𝑡𝐵𝐿}𝑏,𝐿^−𝛼/2𝑓=𝑏−𝑏_2𝐵𝐿^−𝛼/2𝑓−𝐿^−𝛼/2𝑏−𝑏_2𝐵𝑓₁−𝐿^−𝛼/2𝑏−𝑏_2𝐵𝑓₂𝑏,𝐿^−𝛼/2𝑓=𝑒^{−𝑡𝐵𝐿}𝑏−𝑏_2𝐵𝐿^−𝛼/2𝑓−𝐿^−𝛼/2𝑏−𝑏_2𝐵𝑓₁−𝐿^−𝛼/2𝑏−𝑏_2𝐵𝑓₂.(4.15) Then ||𝐵1||_𝐵||𝑏,𝐿^−𝛼/2𝑓(𝑦)−𝑒^{−𝑡𝐵𝐿}𝑏,𝐿^−𝛼/2𝑓(𝑦)||^𝛿𝑑𝑦^1/𝛿≤𝐶1||𝐵||_𝐵||𝑏(𝑦)−𝑏_2𝐵𝐿^−𝛼/2𝑓(𝑦)||^𝛿𝑑𝑦^1/𝛿+𝐶1||𝐵||_𝐵||𝐿^−𝛼/2𝑏−𝑏_2𝐵𝑓₁(𝑦)||^𝛿𝑑𝑦^1/𝛿+𝐶1||𝐵||_𝐵||𝑒^{−𝑡𝐵𝐿}𝑏−𝑏_2𝐵𝐿^−𝛼/2𝑓(𝑦)||^𝛿𝑑𝑦^1/𝛿+𝐶1||𝐵||_𝐵||𝑒^{−𝑡𝐵𝐿}𝐿^−𝛼/2𝑏−𝑏_2𝐵𝑓₁(𝑦)||^𝛿𝑑𝑦^1/𝛿+𝐶1||𝐵||_𝐵||𝐿^−𝛼/2−𝑒^{−𝑡𝐵𝐿}𝐿^−𝛼/2𝑏−𝑏_2𝐵𝑓₂(𝑦)||^𝛿𝑑𝑦^1/𝛿≐𝐼+𝐼𝐼+𝐼𝐼𝐼+𝐼𝑉+𝑉.(4.16)

We are going to estimate each term, respectively. Fix 0<𝛿<1 and choose a real number 𝜏 such that 1<𝜏<2 and 𝜏′𝛿<1. Since 𝜔∈𝐴₁, then it follows from Hölder's inequality that 𝐼≤𝐶1||𝐵||_𝐵||𝑏(𝑦)−𝑏_2𝐵||^𝜏𝛿𝑑𝑦^1/𝜏𝛿_𝐵||𝐿^−𝛼/2𝑓(𝑦)||^𝜏𝛿𝑑𝑦^{1/𝜏𝛿}≤𝐶1||𝐵||_2𝐵≤𝐶‖𝑏‖_{Lip𝛽(𝜔)}||𝑏(𝑦)−𝑏_2𝐵||𝑑𝑦_𝐵||𝐿^−𝛼/2𝑓(𝑦)||𝑑𝑦1||2𝐵||𝜔(2𝐵)^1+𝛽/𝑛_𝐵≤𝐶‖𝑏‖_{Lip𝛽(𝜔)}||𝐿^−𝛼/2𝑓(𝑦)||𝑑𝑦𝜔(𝑥)^1+𝛽/𝑛𝑀_𝛽,1𝐿^−𝛼/2𝑓(𝑥).(4.17)

For 𝐼𝐼, using Hölder's inequality, Kolmogorov's inequality (see page 485 [8],) and Remark 3.6, then we deduce that 𝐼𝐼≤𝐶1||𝐵||_𝐵≤𝐶1||𝐿^−𝛼/2𝑏−𝑏_2𝐵𝑓₁(𝑦)||𝑑𝑦||𝐵||||𝐵||^𝛼/𝑛‖‖𝐿^−𝛼/2𝑏−𝑏_2𝐵𝑓₁‖‖_{𝐿𝑛/(𝑛−𝛼),∞}≤𝐶1||𝐵||^{1−𝛼/𝑛}_𝐵≤𝐶1||𝑏(𝑦)−𝑏_2𝐵𝑓₁(𝑦)||𝑑𝑦||𝐵||^{1−𝛼/𝑛}_2𝐵||𝑓(𝑦)||^𝑟𝑤(𝑦)𝑑𝑦^1/𝑟_2𝐵||𝑏(𝑦)−𝑏_2𝐵||^𝑟𝑤(𝑦)^{−𝑟/𝑟}𝑑𝑦^1/𝑟≤𝐶‖𝑏‖_{Lip𝛽(𝜔)}𝑀_{𝛼+𝛽,𝑟,𝜔}𝑓(𝑥)𝜔(2𝐵)||2𝐵||^{1−𝛼/𝑛}≤𝐶‖𝑏‖_{Lip𝛽(𝜔)}𝜔(𝑥)^{1−𝛼/𝑛}𝑀_{𝛼+𝛽,𝑟,𝜔}𝑓(𝑥),(4.18) where we have used the condition that 𝜔∈𝐴₁.

Using Hölder's inequality and Lemma 4.6, we obtain that 𝐼𝐼𝐼≤𝐶‖𝑏‖_{Lip𝛽(𝜔)}𝜔(𝑥)𝑀_{𝛽,𝑟,𝜔}𝐿^−𝛼/2𝑓(𝑥).(4.19)

For 𝐼𝑉, using the estimate in II, we get 𝐼𝑉≤𝐶||𝐵||_𝐵_2𝐵≤𝐶||𝑝_𝑡𝐵(𝑦,𝑧)||||𝐿^−𝛼/2𝑏−𝑏_2𝐵𝑓(𝑧)||𝑑𝑧𝑑𝑦||2𝐵||_2𝐵≤𝐶‖𝑏‖||𝐿^−𝛼/2_{Lip𝛽(𝜔)}𝑏−𝑏_2𝐵𝑓(𝑧)||𝑑𝑧𝜔(𝑥)^{1−𝛼/𝑛}𝑀_{𝛼+𝛽,𝑟,𝜔}𝑓(𝑥).(4.20)

By virtue of Lemma 3.1, we have the following: 𝑉≤𝐶||𝐵||_𝐵_(2𝐵)𝑐≤𝐶_∞||𝐾_{𝛼,𝑡𝐵}(𝑦,𝑧)||||𝑏(𝑧)−𝑏_2𝐵𝑓(𝑧)||𝑑𝑧𝑑𝑦_𝑘=1_{2𝑘𝑟𝐵≤|𝑥0−𝑧|<2𝑘+1𝑟𝐵}1||𝑥₀−𝑧||^𝑛−𝛼𝑟^2𝐵||𝑥₀−𝑧||²≤𝐶_∞||𝑏(𝑧)−𝑏_2𝐵𝑓(𝑧)||𝑑𝑧_𝑘=12^−2𝑘1||2^𝑘+1𝐵||^{1−𝛼/𝑛}_2𝑘+1𝐵≤𝐶_∞||𝑏(𝑧)−𝑏_2𝐵𝑓(𝑧)||𝑑𝑧_𝑘=12^−2𝑘1||2^𝑘+1𝐵||^{1−𝛼/𝑛}_2𝑘+1𝐵+𝐶_∞||𝑏(𝑧)−𝑏_2𝑘+1𝐵𝑓(𝑧)||𝑑𝑧_𝑘=12^−2𝑘||𝑏_2𝑘+1𝐵−𝑏_2𝐵||1||2^𝑘+1𝐵||^{1−𝛼/𝑛}_2𝑘+1𝐵≐𝑉𝐼+𝑉𝐼𝐼.||𝑓(𝑧)||𝑑𝑧(4.21) Making use of the same argument as that of 𝐼𝐼, we have 𝑉𝐼≤𝐶‖𝑏‖_{Lip𝛽(𝜔)}𝜔(𝑥)^{1−𝛼/𝑛}𝑀_{𝛼+𝛽,𝑟,𝜔}𝑓(𝑥).(4.22)

Note that 𝜔∈𝐴₁, ||𝑏_2𝑘+1𝐵−𝑏_2𝐵||≤𝐶𝑘𝜔(𝑥)‖𝑏‖_{Lip𝛽(𝜔)}𝜔2^𝑘+1𝐵^𝛽/𝑛.(4.23) So, the value of 𝑉𝐼𝐼 can be controlled by 𝐶‖𝑏‖_{Lip𝛽(𝜔)}𝜔(𝑥)^1+𝛽/𝑛𝑀_𝛼+𝛽,1𝑓(𝑥).(4.24)

Combining the above estimates for I–V, we finish the proof of Lemma 4.7.

Proof of Theorem 1.2. We first prove (𝑎). As before, we only prove Theorem 1.2 in the case 0<𝛼<1. For 0<𝛼+𝛽<𝑛 and 1<𝑝<𝑛/(𝛼+𝛽), we can find a number 𝑟 such that 1<𝑟<𝑝. By Lemma 4.7, we obtain the following: ‖‖𝑏,𝐿^−𝛼/2𝑓‖‖_𝐿^{𝑞,𝑘𝑞/𝑝}_{(𝜔1−(1−𝛼/𝑛)𝑞,𝜔)}≤𝐶‖‖𝑀^#𝐿,𝛿𝑏,𝐿^−𝛼/2𝑓‖‖_𝐿^{𝑞,𝑘𝑞/𝑝}_{(𝜔1−(1−𝛼/𝑛)𝑞,𝜔)}≤𝐶‖𝑏‖_{Lip𝛽(𝜔)}‖‖𝜔(⋅)^1+𝛽/𝑛𝑀_𝛽,1𝐿^−𝛼/2𝑓‖‖_{𝐿𝑞,𝑘𝑞/𝑝(𝜔1−(1−𝛼/𝑛)𝑞,𝜔)}+‖‖𝜔(⋅)^{1−𝛼/𝑛}𝑀_{𝛼+𝛽,𝑟,𝜔}𝑓‖‖_{𝐿𝑞,𝑘𝑞/𝑝(𝜔1−(1−𝛼/𝑛)𝑞,𝜔)}≤𝐶‖𝑏‖_{Lip𝛽(𝜔)}+‖‖𝜔(⋅)^1+𝛽/𝑛𝑀_𝛼+𝛽,1𝑓‖‖_{𝐿𝑞,𝑘𝑞/𝑝(𝜔1−(1−𝛼/𝑛)𝑞,𝜔)}‖‖𝑀^𝛽,1𝐿^−𝛼/2𝑓‖‖_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞/𝑝,𝜔)}+‖‖𝑀_{𝛼+𝛽,𝑟,𝜔}𝑓‖‖_{𝐿𝑞,𝑘𝑞/𝑝(𝜔)}+‖‖𝑀_𝛼+𝛽,1𝑓‖‖_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞/𝑝,𝜔)}.(4.25) Let 1/𝑞₁=1/𝑝−𝛼/𝑛 and 1/𝑞=1/𝑞₁−𝛽/𝑛. Lemmas 4.1–4.4 yield that ‖‖𝑏,𝐿^−𝛼/2𝑓‖‖_𝐿^{𝑞,𝑘𝑞/𝑝}_{(𝜔1−(1−𝛼/𝑛)𝑞,𝜔)}≤≤𝐶𝐶‖‖𝑏𝑏‖‖_{Lip𝛽(𝜔)Lip𝛽(𝜔)}‖‖𝐿^−𝛼/2𝑓‖‖_{𝐿𝑞11,𝑘𝑞/𝑝(𝜔𝑞1/𝑝,𝜔)}+‖𝑓‖_𝐿^𝑝,𝑘_(𝜔)‖𝑓‖_𝐿^𝑝,𝑘_(𝜔).(4.26)
Now we prove (b). Let 𝐿=−Δ be the Laplacian on ℝ^𝑛, then 𝐿^−𝛼/2 is the classical fractional integral 𝐼_𝛼. We use the same argument as Janson [13]. Choose 𝑍₀∈ℝ^𝑛 so that |𝑍₀|=3. For 𝑥∈𝐵(𝑍₀,2),|𝑥|^−𝛼+𝑛 can be written as the absolutely convergent Fourier series, |𝑥|^−𝛼+𝑛=∑_{𝑚∈ℤ𝑛}𝑎_𝑚𝑒^{𝑖⟨𝜈𝑚,𝑥⟩} with ∑_𝑚|𝑎_𝑚|<∞ since |𝑥|^−𝛼+𝑛∈𝐶^∞(𝐵(𝑍₀,2)). For any 𝑥₀∈ℝ^𝑛 and 𝜌>0, let 𝐵=𝐵(𝑥₀,𝜌) and 𝐵_𝑍0=𝐵(𝑥₀+𝑍₀𝜌,𝜌), _𝐵|||𝑏(𝑥)−𝑏_𝐵𝑍0|||𝑑𝑥=1||𝐵_𝑍0||_𝐵|||||_𝐵𝑍0=1(𝑏(𝑥)−𝑏(𝑦))𝑑𝑦|||||𝑑𝑥𝜌^𝑛_𝐵𝑠(𝑥)_𝐵𝑍0(𝑏(𝑥)−𝑏(𝑦))||𝑥−𝑦||^−𝛼+𝑛||𝑥−𝑦||^𝑛−𝛼𝑑𝑦𝑑𝑥,(4.27) where 𝑠(𝑥)=sgn(∫_𝐵𝑍0(𝑏(𝑥)−𝑏(𝑦))𝑑𝑦). Fix 𝑥∈𝐵 and 𝑦∈𝐵_𝑍0, then (𝑦−𝑥)/𝜌∈𝐵_𝑍0,2, hence, 𝜌^−𝛼+𝑛𝜌^𝑛_𝐵𝑠(𝑥)_𝐵𝑍0(𝑏(𝑥)−𝑏(𝑦))||𝑥−𝑦||^−𝛼+𝑛||𝑥−𝜌𝑦||^𝑛−𝛼=𝜌_−𝛼𝑑𝑦𝑑𝑥_𝑚_∈𝑍𝑛𝑎_𝑚_𝐵≤𝜌_−𝛼𝑠(𝑥)_𝐵𝑍0(𝑏(𝑥)−𝑏(𝑦))||𝑥−𝑦||^𝑛−𝛼𝑒^{𝑖⟨𝜈𝑚,𝑦/𝜌⟩}𝑑𝑦𝑒^{−𝑖⟨𝜈𝑚,𝑥/𝜌⟩}𝑑𝑥|||||_{𝑚∈𝑍𝑛}||𝑎_𝑚||_𝐵𝑠(𝑥)𝑏,𝐿^−𝛼/2𝜒_𝐵𝑍0𝑒^{𝑖⟨𝜈𝑚,⋅/𝜌⟩}≤𝜌_−𝛼(𝑥)𝜒^𝐵(𝑥)𝑒^{−𝑖⟨𝜈𝑚,𝑥/𝜌⟩}𝑑𝑥|||||_𝑚_∈𝑍𝑛||𝑎_𝑚||‖‖𝑏,𝐿^−𝛼/2𝜒_𝐵𝑍0𝑒^{𝑖⟨𝜈𝑚,⋅/𝜌⟩}‖‖_{𝐿𝑞,0(𝜔1−(1−𝛼/𝑛)𝑞,𝜔)}_𝐵𝜔(𝑥)^{𝑞′(1/𝑞′−𝛼/𝑛)}𝑑𝑥^1/𝑞≤𝐶𝜌^−𝛼_𝑚_∈𝑍𝑛||𝑎_𝑚||‖‖𝜒_𝐵𝑍0‖‖_{𝐿𝑝,0(𝜔)}_𝐵𝜔(𝑥)^{𝑞′(1/𝑞′−𝛼/𝑛)}𝑑𝑥^1/𝑞≤𝐶𝜔(𝐵)^{1/𝑝+1/𝑞′−𝛼/𝑛}=𝐶𝜔(𝐵)^1+𝛽/𝑛.(4.28) This implies that 𝑏∈Lip_𝛽(𝜔). Thus, (b) is proved.

Acknowledgment

The authors thank the referee for the useful suggestions. Z. Si was supported by Doctoral Foundation of Henan Polytechnic University. F. Zhao was supported by Shanghai Leading Academic Discipline Project (Grant no. J50101).