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Abstract and Applied Analysis
VolumeΒ 2012, Article IDΒ 929381, 14 pages
http://dx.doi.org/10.1155/2012/929381
Research Article

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

1School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China
2Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 20 March 2012; Revised 5 July 2012; Accepted 20 July 2012

Academic Editor: GiovanniΒ Galdi

Copyright Β© 2012 Zengyan Si and Fayou Zhao. 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

We prove that 𝑏 is in Lip𝛽(πœ”) if and only if the commutator [𝑏,πΏβˆ’π›Ό/2] of the multiplication operator by 𝑏 and the general fractional integral operator πΏβˆ’π›Ό/2 is bounded from the weighted Morrey space 𝐿𝑝,π‘˜(πœ”) to πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”1βˆ’(1βˆ’π›Ό/𝑛)π‘ž,πœ”), where 0<𝛽<1, 0<𝛼+𝛽<𝑛,1<𝑝<𝑛/(𝛼+𝛽), 1/π‘ž=1/π‘βˆ’(𝛼+𝛽)/𝑛, 0β‰€π‘˜<𝑝/π‘ž, πœ”π‘ž/π‘βˆˆπ΄1, and π‘Ÿπœ”>(1βˆ’π‘˜)/(𝑝/(π‘žβˆ’π‘˜)), and here π‘Ÿπœ” denotes the critical index of πœ” for the reverse HΓΆlder condition.

1. Introduction and Main Results

Suppose that 𝐿 is a linear operator on 𝐿2(ℝ𝑛) which generates an analytic semigroup π‘’βˆ’π‘‘πΏ with a kernel 𝑝𝑑(π‘₯,𝑦) satisfying a Gaussian upper bound, that is, ||𝑝𝑑(||≀𝐢π‘₯,𝑦)𝑑𝑛/2π‘’βˆ’π‘(|π‘₯βˆ’π‘¦|2/𝑑)(1.1) for π‘₯,π‘¦βˆˆβ„π‘› and all 𝑑>0. Since we assume only upper bound on heat kernel 𝑝𝑑(π‘₯,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 πΏβˆ’π›Ό/2 of the operator 𝐿 is defined by πΏβˆ’π›Ό/21𝑓(π‘₯)=ξ€œΞ“(𝛼/2)∞0π‘’βˆ’π‘‘πΏπ‘“π‘‘π‘‘π‘‘βˆ’π›Ό/2+1(π‘₯).(1.2)

Note that, if 𝐿=βˆ’Ξ” is the Laplacian on ℝ𝑛, then, πΏβˆ’π›Ό/2 is the classical fractional integral 𝐼𝛼 which plays important roles in many fields. Let 𝑏 be a locally integrable function on ℝ𝑛, the commutator of 𝑏 and πΏβˆ’π›Ό/2 is defined by 𝑏,πΏβˆ’π›Ό/2𝑓(π‘₯)=𝑏(π‘₯)πΏβˆ’π›Ό/2𝑓(π‘₯)βˆ’πΏβˆ’π›Ό/2(𝑏𝑓)(π‘₯).(1.3)

For the special case of 𝐿=βˆ’Ξ”, many results have been produced. PaluszyΕ„ski [2] obtained that π‘βˆˆLip𝛽(ℝ𝑛) if the commutator [𝑏,𝐼𝛼] is bounded from 𝐿𝑝(ℝ𝑛) to πΏπ‘Ÿ(ℝ𝑛), where 1<𝑝<π‘Ÿ<∞,0<𝛽<1 and 1/π‘βˆ’1/π‘Ÿ=(𝛼+𝛽)/𝑛 with 𝑝<𝑛/(𝛼+𝛽). Shirai [3] proved that π‘βˆˆLip𝛽(ℝ𝑛) if and only if the commutator [𝑏,𝐼𝛼] is bounded from the classical Morrey spaces 𝐿𝑝,πœ†(ℝ𝑛) to πΏπ‘ž,πœ†(ℝ𝑛) for 1<𝑝<π‘ž<∞,0<𝛼,0<𝛽<1, and 0<𝛼+𝛽=(1/π‘βˆ’1/π‘ž)(π‘›βˆ’πœ†)<𝑛 or 𝐿𝑝,πœ†(ℝ𝑛) to πΏπ‘ž,πœ‡(ℝ𝑛) for 1<𝑝<π‘ž<∞,0<𝛼,0<𝛽<1,0<𝛼+𝛽=(1/π‘βˆ’1/π‘ž)<𝑛,0<πœ†<π‘›βˆ’(𝛼+𝛽)𝑝, and πœ‡/π‘ž=πœ†/𝑝. Wang [4] established some weighted boundedness of properties of commutator [𝑏,𝐼𝛼] on the weighted Morrey spaces 𝐿𝑝,π‘˜ under appropriated conditions on the weight πœ”, 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 π‘βˆˆLip𝛽(ℝ𝑛), then the commutator [𝑏,πΏβˆ’π›Ό/2] is bounded from 𝐿𝑝(πœ”π‘) to πΏπ‘ž(πœ”π‘ž), where 0<𝛽<1,0<𝛼+𝛽<𝑛,1<𝑝<𝑛/(𝛼+𝛽),1/π‘βˆ’1/π‘ž=(𝛼+𝛽)/𝑛, and πœ”π‘žβˆˆπ΄1.

The purpose of this paper is to give necessary and sufficient conditions for boundedness of commutators of the general fractional integrals with π‘βˆˆLip𝛽(πœ”) (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/π‘ž=1/π‘βˆ’(𝛼+𝛽)/𝑛,0β‰€π‘˜<min{𝑝/π‘ž,𝑝𝛽/𝑛}, and πœ”π‘žβˆˆπ΄1. Then one has the following. (a)If π‘βˆˆLip𝛽(ℝ𝑛), then [𝑏,πΏβˆ’π›Ό/2] is bounded from 𝐿𝑝,π‘˜(πœ”π‘,πœ”π‘ž) to πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž); (b)If [𝑏,πΏβˆ’π›Ό/2] is bounded from 𝐿𝑝,π‘˜(πœ”π‘,πœ”π‘ž) to πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž), then π‘βˆˆLip𝛽(ℝ𝑛).

Theorem 1.2. Let 0<𝛽<1, 0<𝛼+𝛽<𝑛,1<𝑝<𝑛/(𝛼+𝛽), 1/π‘ž=1/π‘βˆ’(𝛼+𝛽)/𝑛,0β‰€π‘˜<𝑝/π‘ž,πœ”π‘ž/π‘βˆˆπ΄1, and π‘Ÿπœ”>(1βˆ’π‘˜)/(𝑝/(π‘žβˆ’π‘˜)), where π‘Ÿπœ” denotes the critical index of πœ” for the reverse HΓΆlder condition. Then one has the following. (a)If π‘βˆˆLip𝛽(πœ”), then [𝑏,πΏβˆ’π›Ό/2] is bounded from 𝐿𝑝,π‘˜(πœ”) to πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”1βˆ’(1βˆ’π›Ό/𝑛)π‘ž,πœ”); (b)If [𝑏,πΏβˆ’π›Ό/2] is bounded from 𝐿𝑝,π‘˜(πœ”) to πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”1βˆ’(1βˆ’π›Ό/𝑛)π‘ž,πœ”), then π‘βˆˆLip𝛽(πœ”).

Our results not only extend the results of [4] from (βˆ’Ξ”) to a general operator 𝐿, but also characterize the (weighted) Lipschitz spaces by means of the boundedness of [𝑏,πΏβˆ’π›Ό/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 𝐡 by |𝐡| and the weighted measure of 𝐡 by πœ”(𝐡), where βˆ«πœ”(𝐡)=π΅πœ”(π‘₯)𝑑π‘₯. For a measurable set 𝐸, denote by πœ’πΈ 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 𝐢 will be used for various constants, and may change from one occurrence to another.

2. Some Preliminaries

A nonnegative function πœ” defined on ℝ𝑛 is called weight if it is locally integral. A weight πœ” is said to belong to the Muckenhoupt class 𝐴𝑝(ℝ𝑛) for 1<𝑝<∞, if there exists a positive constant 𝐢 such that ξ‚΅1||𝐡||ξ€œπ΅1πœ”(π‘₯)𝑑π‘₯ξ‚Άξ‚΅||𝐡||ξ€œπ΅πœ”(π‘₯)βˆ’1/(π‘βˆ’1)𝑑π‘₯π‘βˆ’1≀𝐢,(2.1) for every ball π΅βŠ‚β„π‘›. The class 𝐴1(ℝ𝑛) is defined replacing the above inequality by ξ‚΅1||𝐡||ξ€œπ΅ξ‚Άπœ”(π‘₯)𝑑π‘₯≀𝐢essinfπ‘₯βˆˆπ΅πœ”(π‘₯).(2.2) When 𝑝=∞,πœ”βˆˆπ΄βˆž, if there exist positive constants 𝛿 and 𝐢 such that given a ball 𝐡 and 𝐸 is a measurable subset of 𝐡, then πœ”(𝐸)ξ‚΅||𝐸||πœ”(𝐡)≀𝐢||𝐡||𝛿.(2.3)

A weight function πœ” belongs to 𝐴𝑝,π‘ž for 1<𝑝<π‘ž<∞ if for every ball 𝐡 in ℝ𝑛, there exists a positive constant 𝐢 which is independent of 𝐡 such that ξ‚΅1ξ€œ|𝐡|π΅πœ”(π‘₯)π‘žξ‚Άπ‘‘π‘₯1/π‘žξ‚΅1ξ€œ|𝐡|π΅πœ”(π‘₯)βˆ’π‘β€²ξ‚Άπ‘‘π‘₯1/𝑝′≀𝐢.(2.4)

From the definition of 𝐴𝑝,π‘ž, we can get that πœ”βˆˆπ΄π‘,π‘ž,iο¬€πœ”π‘žβˆˆπ΄1+π‘ž/𝑝′.(2.5) Since πœ”π‘ž/π‘βˆˆπ΄1, then by (2.5), we have πœ”1/π‘βˆˆπ΄π‘,π‘ž.

A weight function πœ” belongs to the reverse HΓΆlder class RHπ‘Ÿ if there exist two constants π‘Ÿ>1 and 𝐢>0 such that the following reverse HΓΆlder inequality, ξ‚΅1ξ€œ|𝐡|π΅πœ”(π‘₯)π‘Ÿξ‚Άπ‘‘π‘₯1/π‘Ÿ1≀𝐢||𝐡||ξ€œπ΅πœ”(π‘₯)𝑑π‘₯,(2.6) holds for every ball 𝐡 in ℝ𝑛.

It is well known that if πœ”βˆˆπ΄π‘ with 1≀𝑝<∞, then there exists π‘Ÿ>1 such that πœ”βˆˆRHπ‘Ÿ. It follows from HΓΆlder inequality that πœ”βˆˆRHπ‘Ÿ implies πœ”βˆˆRH𝑠 for all 1<𝑠<π‘Ÿ. Moreover, if πœ”βˆˆRHπ‘Ÿ,π‘Ÿ>1, then we have πœ”βˆˆRHπ‘Ÿ+πœ€ for some πœ–>0. We thus write π‘Ÿπ‘€=sup{π‘Ÿ>1βˆΆπœ”βˆˆRHπ‘Ÿ} to denote the critical index of πœ” for the reverse HΓΆlder condition. For more details on Muckenhoupt class 𝐴𝑝,π‘ž, we refer the reader to [8–10].

Definition 2.1 (see [5]). Let 1≀𝑝<∞ and 0β‰€π‘˜<1. Then for two weights πœ‡ and 𝜈, the weighted Morrey space is defined by 𝐿𝑝,π‘˜ξ€½(πœ‡,𝜈)=π‘“βˆˆπΏπ‘loc(πœ‡)βˆΆβ€–π‘“β€–πΏπ‘,π‘˜(πœ‡,𝜈)ξ€Ύ,<∞(2.7) where ‖𝑓‖𝐿𝑝,π‘˜(πœ‡,𝜈)=sup𝐡1𝜈(𝐡)π‘˜ξ€œπ΅||||𝑓(π‘₯)π‘ξ‚Άπœ‡(π‘₯)𝑑π‘₯1/𝑝,(2.8) and the supremum is taken over all balls 𝐡 in ℝ𝑛.

If πœ‡=𝜈, then we have the classical Morrey space 𝐿𝑝,π‘˜(πœ‡) with measure πœ‡. When π‘˜=0, then 𝐿𝑝,0(πœ‡)=𝐿𝑝(πœ‡) is the Lebesgue space with measure πœ‡.

Definition 2.2 (see [11]). Let 1≀𝑝<∞, 0<𝛽<1, and πœ”βˆˆπ΄βˆž. A locally integral function 𝑏 is said to be in Lip𝑝𝛽(πœ”) if ‖𝑏‖Lip𝑝𝛽(πœ”)=sup𝐡1πœ”(𝐡)𝛽/𝑛1ξ€œπœ”(𝐡)𝐡||𝑏(π‘₯)βˆ’π‘π΅||π‘πœ”(π‘₯)1βˆ’π‘ξ‚Άπ‘‘π‘₯1/𝑝≀𝐢<∞,(2.9) where 𝑏𝐡=|𝐡|βˆ’1βˆ«π΅π‘(𝑦)𝑑𝑦 and the supremum is taken over all ball π΅βŠ‚π‘…π‘›. When 𝑝=1, we denote Lip𝑝𝛽(πœ”) by Lip𝛽(πœ”).

Obviously, for the case πœ”=1, then the Lip𝑝𝛽(πœ”) space is the classical Lip𝑝𝛽 space.

Remark 2.3. Let πœ”βˆˆπ΄1, GarcΓ­a-Cuerva [11] proved that the spaces ‖𝑓‖Lip𝑝𝛽(πœ”) coincide, and the norms of ||β‹…||Lip𝑝𝛽(πœ”) 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 𝑀𝛽,π‘Ÿπ‘“(π‘₯)=supπ‘₯βˆˆπ΅ξƒ©1||𝐡||1βˆ’π›½π‘Ÿ/π‘›ξ€œπ΅||||𝑓(𝑦)π‘Ÿξƒͺ𝑑𝑦1/π‘Ÿ,π‘Ÿβ‰₯1,(2.10) when 0<𝛽<𝑛. If 𝛽=0 and π‘Ÿ=1, then 𝑀0,1𝑓=𝑀𝑓 denotes the usual Hardy-Littlewood maximal function.

Let πœ” be a weight. The weighted maximal operator π‘€πœ” is defined by π‘€πœ”π‘“(π‘₯)=supπ‘₯∈𝐡1ξ€œπœ”(𝐡)𝐡||||𝑓(𝑦)𝑑𝑦.(2.11) The fractional weighted maximal operator 𝑀𝛽,π‘Ÿ,πœ” is defined by 𝑀𝛽,π‘Ÿ,πœ”π‘“(π‘₯)=supπ‘₯βˆˆπ΅ξ‚΅1πœ”(𝐡)1βˆ’π›½π‘Ÿ/π‘›ξ€œπ΅||||𝑓(𝑦)π‘Ÿξ‚Άπœ”(𝑦)𝑑𝑦1/π‘Ÿ,(2.12) where 0≀𝛽<𝑛 and π‘Ÿβ‰₯1. For any π‘“βˆˆπΏπ‘(ℝ𝑛),𝑝β‰₯1, the sharp maximal function 𝑀#𝐿𝑓 associated the generalized approximations to the identity {π‘’βˆ’π‘‘πΏ,𝑑>0} is given by Martell [7] as follows: 𝑀#𝐿𝑓(π‘₯)=supπ‘₯∈𝐡1||𝐡||ξ€œπ΅||𝑓(𝑦)βˆ’π‘’βˆ’π‘‘π΅πΏ||𝑓(𝑦)𝑑𝑦,(2.13) where 𝑑𝐡=π‘Ÿ2𝐡 and π‘Ÿπ΅ is the radius of the ball 𝐡. 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 𝑀𝛿𝑓=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 π‘’βˆ’π‘‘πΏ has a kernel 𝑝𝑑(π‘₯,𝑦) which satisfies the upper bound (1.1). Let πœ†>0 and π‘“βˆˆπΏπ‘(ℝ𝑛) for some 1<𝑝<∞. Suppose that πœ”βˆˆπ΄βˆž, then for every 0<πœ‚<1, there exists a real number 𝛾>0 independent of 𝛾,𝑓 such that one has the following weighted version of the local good πœ† inequality, for πœ‚>0, 𝐴>1, πœ”ξ€½π‘₯βˆˆβ„π‘›βˆΆπ‘€π›Ώπ‘“>π΄πœ†,𝑀#𝐿,𝛿𝑓(π‘₯)β‰€π›Ύπœ†β‰€πœ‚πœ”π‘₯βˆˆβ„π‘›βˆΆπ‘€π›Ώπ‘“ξ€Ύ,(π‘₯)>πœ†(2.15) where 𝐴>1 is a fixed constant which depends only on 𝑛.

If πœ‡,𝜈∈𝐴∞,1<𝑝<∞,0β‰€π‘˜<1, then ‖𝑓‖𝐿𝑝,π‘˜(πœ‡,𝜈)≀‖‖𝑀𝛿𝑓‖‖𝐿𝑝,π‘˜(πœ‡,𝜈)‖‖𝑀≀𝐢#𝐿,𝛿𝑓‖‖𝐿𝑝,π‘˜(πœ‡,𝜈).(2.16) In particular, when πœ‡=𝜈=πœ” and πœ”βˆˆπ΄βˆž, we have ‖𝑓‖𝐿𝑝,π‘˜(πœ”)≀‖‖𝑀𝛿𝑓‖‖𝐿𝑝,π‘˜(πœ”)‖‖𝑀≀𝐢#𝐿,𝛿𝑓‖‖𝐿𝑝,π‘˜(πœ”).(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 π‘’βˆ’π‘‘πΏ has a kernel 𝑝𝑑(π‘₯,𝑦) which satisfies the upper bound (1.1). Then for 0<𝛼<1, the difference operator πΏβˆ’π›Ό/2βˆ’π‘’βˆ’π‘‘πΏπΏβˆ’π›Ό/2 has an associated kernel 𝐾𝛼,𝑑(π‘₯,𝑦) which satisfies 𝐾𝛼,𝑑(𝐢π‘₯,𝑦)≀||||π‘₯βˆ’π‘¦π‘›βˆ’π›Όπ‘‘||||π‘₯βˆ’π‘¦2.(3.1)

Lemma 3.2 (see [4]). Let 0<𝛼+𝛽<𝑛, 1<𝑝<𝑛/(𝛼+𝛽),1/π‘ž=1/π‘βˆ’(𝛼+𝛽)/𝑛, and πœ”βˆˆπ΄1. Then for every 0<π‘˜<𝑝/π‘ž and 1<π‘Ÿ<𝑝, one has ‖‖𝑀𝛼+𝛽,π‘Ÿπ‘“β€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž)≀𝐢‖𝑓‖𝐿𝑝,π‘ž(πœ”π‘,πœ”π‘ž).(3.2)

Lemma 3.3 (see [5]). Let 0<𝛽<𝑛, 1<𝑝<𝑛/𝛽,1/𝑠=1/π‘βˆ’π›½/𝑛, and πœ”βˆˆπ΄π‘,𝑠. Then for every 0<π‘˜<𝑝/𝑠, one has ‖‖𝑀𝛽,1𝑓‖‖𝐿𝑠,π‘˜π‘ /𝑝(πœ”π‘ )≀𝐢‖𝑓‖𝐿𝑝,π‘˜(πœ”π‘,πœ”π‘ ).(3.3)

Lemma 3.4 (see [4]). Let 0<𝛼+𝛽<𝑛, 1<𝑝<𝑛/(𝛼+𝛽),1/π‘ž=1/π‘βˆ’π›Ό/𝑛, 1/𝑠=1/π‘žβˆ’π›½/𝑛, and πœ”π‘žβˆˆπ΄1. Then for every 0<π‘˜<𝑝/𝑠, one has ‖‖𝑀𝛽,1𝑓‖‖𝐿𝑠,π‘˜π‘ /𝑝(πœ”π‘ )β‰€πΆβ€–π‘“β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž,πœ”π‘ ).(3.4)

Lemma 3.5. Let 0<𝛼+𝛽<𝑛, 1<𝑝<𝑛/(𝛼+𝛽),1/π‘ž=1/π‘βˆ’π›Ό/𝑛, 1/𝑠=1/π‘žβˆ’π›½/𝑛, and πœ”π‘žβˆˆπ΄1. Then for every 0<π‘˜<𝑝𝛽/𝑛, one has β€–β€–πΏβˆ’π›Ό/2π‘“β€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž,πœ”π‘ )≀𝐢‖𝑓‖𝐿𝑝,π‘˜(πœ”π‘,πœ”π‘ ).(3.5)

Proof. Since the semigroup π‘’βˆ’π‘‘πΏ has a kernel 𝑝𝑑(π‘₯,𝑦) which satisfies the upper bound (1.1), it is easy to check that πΏβˆ’π›Ό/2𝑓(π‘₯)≀𝐢𝐼𝛼(|𝑓|)(π‘₯) for all π‘₯βˆˆβ„π‘›. Together with the result (cf. [4]), that is, β€–β€–πΌπ›Όπ‘“β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž,πœ”π‘ )‖‖≀𝐢𝑓‖𝐿𝑝,π‘˜(πœ”π‘,πœ”π‘ ),(3.6) we can get the desired result.

Remark 3.6. Using the boundedness property of 𝐼𝛼, we also know πΏβˆ’π›Ό/2 is bounded from 𝐿1 to weak 𝐿𝑛/(π‘›βˆ’π›Ό). It is easy to check that Lemmas 3.2–3.5 also hold when π‘˜=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 π‘βˆˆLip𝛽(ℝ𝑛). Then for all π‘Ÿ>1 and for all π‘₯βˆˆβ„π‘›, one has 𝑀#𝐿,𝛿𝑏,πΏβˆ’π›Ό/2𝑓(π‘₯)≀𝐢‖𝑏‖Lip𝛽(ℝ𝑛)𝑀𝛽,1ξ€·πΏβˆ’π›Ό/2𝑓(π‘₯)+𝑀𝛼+𝛽,π‘Ÿπ‘“(π‘₯)+𝑀𝛼+𝛽,1ξ€Έ.𝑓(π‘₯)(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 π‘Ÿ such that 1<π‘Ÿ<𝑝. By (2.17) and Lemma 3.7, we obtain the following: ‖‖𝑏,πΏβˆ’π›Ό/2ξ€»π‘“β€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž)‖‖𝑀≀𝐢#𝐿,𝛿𝑏,πΏβˆ’π›Ό/2ξ€»π‘“ξ€Έβ€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž)≀𝐢‖𝑏‖Lip𝛽(ℝ𝑛)‖‖𝑀𝛽,1ξ€·πΏβˆ’π›Ό/2π‘“ξ€Έβ€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž)+‖‖𝑀𝛼+𝛽,π‘Ÿπ‘“β€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž)+‖‖𝑀𝛼+𝛽,1π‘“β€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž).(3.8) Let 1/π‘ž1=1/π‘βˆ’π›Ό/𝑛 and 1/π‘ž=1/π‘ž1βˆ’π›½/𝑛. Since πœ”π‘žβˆˆπ΄1, then by (2.5), we have πœ”βˆˆπ΄π‘,π‘ž. Since 0<π‘˜<min{𝑝/π‘ž,𝑝𝛽/𝑛}, by Lemmas 3.2–3.5, we yield that ‖‖𝑏,πΏβˆ’π›Ό/2ξ€»π‘“β€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž)≀𝐢‖𝑏‖Lip𝛽(ℝ𝑛)ξ‚€β€–β€–πΏβˆ’π›Ό/2π‘“β€–β€–πΏπ‘ž11,π‘˜π‘ž/𝑝(πœ”π‘ž1,πœ”π‘ž)+‖𝑓‖𝐿𝑝,π‘˜(πœ”π‘,πœ”π‘ž)≀𝐢‖𝑏‖Lip𝛽(ℝ𝑛)‖𝑓‖𝐿𝑝,π‘˜(πœ”π‘,πœ”π‘ž).(3.9)
Now we prove (𝑏). Let 𝐿=βˆ’Ξ” be the Laplacian on ℝ𝑛, then πΏβˆ’π›Ό/2 is the classical fractional integral 𝐼𝛼. Let π‘˜=0 and weight πœ”β‰‘1, then 𝐿𝑝,π‘˜(πœ”π‘,πœ”π‘ž)=𝐿𝑝 and πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž,πœ”)=πΏπ‘ž. From [2], the (𝐿𝑝,πΏπ‘ž) boundedness of [𝑏,𝐼𝛼] implies that π‘βˆˆLip𝛽(ℝ𝑛).
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/π‘ž=1/π‘βˆ’π›Ό/𝑛,1/𝑠=1/π‘žβˆ’π›½/𝑛, and πœ”π‘ /π‘βˆˆπ΄1. Then if 0<π‘˜<𝑝/𝑠 and π‘Ÿπœ”>1/(𝑝/(π‘žβˆ’π‘˜)), one has ‖‖𝑀𝛽,1𝑓‖‖𝐿𝑠,π‘˜π‘ /𝑝(πœ”π‘ /𝑝,πœ”)β‰€πΆβ€–π‘“β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž/𝑝,πœ”).(4.1)

Lemma 4.2 (see [4]). Let 0<𝛼<𝑛,1<𝑝<𝑛/𝛼,1/π‘ž=1/π‘βˆ’π›Ό/𝑛, and πœ”π‘ž/π‘βˆˆπ΄1. Then if 0<π‘˜<𝑝/π‘ž and π‘Ÿπœ”>(1βˆ’π‘˜)/(𝑝/(π‘žβˆ’π‘˜)), one has ‖‖𝑀𝛼,1π‘“β€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž/𝑝,πœ”)≀𝐢‖𝑓‖𝐿𝑝,π‘˜(πœ”).(4.2)

Lemma 4.3 (see [4]). Let 0<𝛼<𝑛,1<𝑝<𝑛/𝛼,1/π‘ž=1/π‘βˆ’π›Ό/𝑛, 0<π‘˜<𝑝/π‘ž, and πœ”βˆˆπ΄βˆž. For any 1<π‘Ÿ<𝑝, one has ‖‖𝑀𝛼,π‘Ÿ,πœ”π‘“β€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž/𝑝,πœ”)≀𝐢‖𝑓‖𝐿𝑝,π‘˜(πœ”).(4.3)

Lemma 4.4. Let 0<𝛼<𝑛,1<𝑝<𝑛/𝛼,1/π‘ž=1/π‘βˆ’π›Ό/𝑛, and πœ”π‘ž/π‘βˆˆπ΄1. Then if 0<π‘˜<𝑝/π‘ž and π‘Ÿπœ”>(1βˆ’π‘˜)/(𝑝/(π‘žβˆ’π‘˜)), one has β€–β€–πΏβˆ’π›Ό/2π‘“β€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž/𝑝,πœ”)≀𝐢‖𝑓‖𝐿𝑝,π‘˜(πœ”).(4.4)

Proof. As before, we know that πΏβˆ’π›Ό/2𝑓(π‘₯)≀𝐢𝐼𝛼(|𝑓|)(π‘₯) for all π‘₯βˆˆβ„π‘›. Using the boundedness property of 𝐼𝛼 on weighted Morrey space (cf. [4]), we have β€–β€–πΏβˆ’π›Ό/2π‘“β€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž/𝑝,πœ”)β‰€β€–β€–πΌπ›Όπ‘“β€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž/𝑝,πœ”)≀𝐢‖𝑓‖𝐿𝑝,π‘˜(πœ”),(4.5) where 1<𝑝<𝑛/𝛼 and 1/π‘ž=1/π‘βˆ’π›Ό/𝑛.

Remark 4.5. It is easy to check that the above lemmas also hold for π‘˜=0.

Lemma 4.6. Assume that the semigroup π‘’βˆ’π‘‘πΏ has a kernel 𝑝𝑑(π‘₯,𝑦) which satisfies the upper bound (1.1), and let π‘βˆˆLip𝛽(πœ”),πœ”βˆˆπ΄1. Then, for every function π‘“βˆˆπΏπ‘(ℝ𝑛),𝑝>1,π‘₯βˆˆβ„π‘›, and 1<π‘Ÿ<∞, one has supπ‘₯∈𝐡1||𝐡||ξ€œπ΅||π‘’βˆ’π‘‘π΅πΏξ€·π‘(𝑦)βˆ’π‘2𝐡||𝑓(𝑦)𝑑𝑦≀𝐢‖𝑏‖Lip𝛽(πœ”)πœ”(π‘₯)𝑀𝛽,π‘Ÿ,πœ”π‘“(π‘₯).(4.6)

Proof. Fix π‘“βˆˆπΏπ‘(ℝ𝑛),1<𝑝<∞ and π‘₯∈𝐡. Then, 1||𝐡||ξ€œπ΅||π‘’βˆ’π‘‘π΅πΏξ€·ξ€·π‘βˆ’π‘2𝐡𝑓(||≀1𝑦)𝑑𝑦||𝐡||ξ€œπ΅ξ€œβ„π‘›||𝑝𝑑𝐡||||ξ€·(𝑦,𝑧)𝑏(𝑧)βˆ’π‘2𝐡||≀1𝑓(𝑧)𝑑𝑧𝑑𝑦||𝐡||ξ€œπ΅ξ€œ2𝐡||𝑝𝑑𝐡||||𝑏(𝑦,𝑧)(𝑧)βˆ’π‘2𝐡𝑓||+1(𝑧)𝑑𝑧𝑑𝑦||𝐡||ξ€œπ΅βˆžξ“π‘˜=1ξ€œ2π‘˜+1𝐡⧡2π‘˜π΅||𝑝𝑑𝐡||||ξ€·(𝑦,𝑧)𝑏(𝑧)βˆ’π‘2𝐡||𝑓(𝑧)𝑑𝑧𝑑𝑦≐ℳ+𝒩.(4.7) It follows from π‘¦βˆˆπ΅ and π‘§βˆˆ2𝐡 that ||𝑝𝑑𝐡||(𝑦,𝑧)β‰€πΆπ‘‘π΅βˆ’π‘›/21≀𝐢||||.2𝐡(4.8) Thus, HΓΆlder's inequality and Definition 2.2 lead to the following: 1ℳ≀𝐢||||ξ€œ2𝐡2𝐡||𝑏(𝑧)βˆ’π‘2𝐡||1𝑓(𝑧)𝑑𝑧≀𝐢||||ξ‚΅ξ€œ2𝐡2𝐡||𝑏(𝑧)βˆ’π‘2𝐡||π‘Ÿξ…žπœ”(𝑧)1βˆ’π‘Ÿξ…žξ‚Άπ‘‘π‘§1/π‘Ÿξ…žξ‚΅ξ€œ2𝐡||||𝑓(𝑧)π‘Ÿξ‚Άπœ”(𝑧)𝑑𝑧1/π‘Ÿβ‰€πΆβ€–π‘β€–Lip𝛽(πœ”)1||||2π΅πœ”(2𝐡)𝛽/𝑛+1/π‘Ÿβ€²πœ”(2𝐡)1/π‘Ÿξ‚΅1πœ”ξ€œ(2𝐡)2𝐡||||𝑓(𝑧)π‘Ÿξ‚Άπœ”(𝑧)𝑑𝑧1/π‘Ÿβ‰€πΆβ€–π‘β€–Lip𝛽(πœ”)1||||2π΅πœ”(2𝐡)𝛽/𝑛+1ξ‚΅1ξ€œπœ”(2𝐡)2𝐡||||𝑓(𝑧)π‘Ÿξ‚Άπœ”(𝑧)𝑑𝑧1/π‘Ÿβ‰€πΆβ€–π‘β€–Lip𝛽(πœ”)πœ”ξ‚΅1(π‘₯)πœ”(2𝐡)1βˆ’π›½π‘Ÿ/π‘›ξ€œ2𝐡||𝑓||(𝑧)π‘Ÿπœ”ξ‚Ά(𝑧)𝑑𝑧1/π‘Ÿβ‰€πΆβ€–π‘β€–Lip𝛽(πœ”)πœ”(π‘₯)𝑀𝛽,π‘Ÿ,πœ”π‘“(π‘₯).(4.9) Moreover, for any π‘¦βˆˆπ΅ and π‘§βˆˆ2π‘˜+1𝐡⧡2π‘˜π΅, we have |π‘¦βˆ’π‘§|β‰₯2π‘˜βˆ’1π‘Ÿπ΅ and |𝑝𝑑𝐡(𝑦,𝑧)|≀𝐢(π‘’βˆ’π‘22(π‘˜βˆ’1)2(π‘˜+1)𝑛/|2π‘˜+1𝐡|)1𝒩=||𝐡||ξ€œπ΅βˆžξ“π‘˜=1ξ€œ2π‘˜+1𝐡⧡2π‘˜π΅||𝑝𝑑𝐡(||||𝑦,𝑧)𝑏(𝑧)βˆ’π‘2𝐡||𝑓(𝑧)π‘‘π‘§π‘‘π‘¦β‰€πΆβˆžξ“π‘˜=1π‘’βˆ’π‘22(π‘˜βˆ’1)2(π‘˜+1)𝑛||2π‘˜+1𝐡||ξ€œ2π‘˜+1𝐡||𝑏(𝑧)βˆ’π‘2𝐡||𝑓(𝑧)π‘‘π‘§β‰€πΆβˆžξ“π‘˜=1π‘’βˆ’π‘22(π‘˜βˆ’1)2(π‘˜+1)𝑛||2π‘˜+1𝐡||ξ€œ2π‘˜+1𝐡||𝑏(𝑧)βˆ’π‘2π‘˜+1𝐡||𝑓(𝑧)𝑑𝑧+πΆβˆžξ“π‘˜=1π‘’βˆ’π‘22(π‘˜βˆ’1)2(π‘˜+1)𝑛||2π‘˜+1𝐡||ξ€œ2π‘˜+1𝐡||𝑏2π‘˜+1π΅βˆ’π‘2𝐡||𝑓(𝑧)𝑑𝑧≐𝒩1+𝒩2.(4.10) We will estimate the values of terms 𝒩1 and 𝒩2, respectively.
Using HΓΆlder's inequality and Remark 2.3, we have the following: 𝒩1β‰€πΆβˆžξ“π‘˜=1π‘’βˆ’π‘22(π‘˜βˆ’1)2(π‘˜+1)𝑛||2π‘˜+1𝐡||Γ—ξ‚΅ξ€œ2π‘˜+1𝐡||𝑏(𝑧)βˆ’π‘2π‘˜+1𝐡||π‘Ÿξ…žπœ”(𝑧)1βˆ’π‘Ÿξ…žξ‚Άπ‘‘π‘§1/π‘Ÿξ…žξ‚΅ξ€œ2π‘˜+1𝐡||||𝑓(𝑧)π‘Ÿξ‚Άπœ”(𝑧)𝑑𝑧1/π‘Ÿβ‰€πΆβˆžξ“π‘˜=12(π‘˜+1)π‘›π‘’βˆ’π‘22(π‘˜βˆ’1)×‖𝑏‖Lip𝛽(πœ”)πœ”ξ€·2π‘˜+1𝐡||2π‘˜+1𝐡||1πœ”ξ€·2π‘˜+1𝐡1βˆ’π›½π‘Ÿ/π‘›ξ€œ2π‘˜+1𝐡||||𝑓(𝑧)π‘Ÿξƒͺπœ”(𝑧)𝑑𝑧1/π‘Ÿβ‰€πΆβ€–π‘β€–Lip𝛽(πœ”)πœ”(π‘₯)𝑀𝛽,π‘Ÿ,πœ”π‘“(π‘₯).(4.11)
By a simple calculation, we have ||𝑏2π‘˜+1π΅βˆ’π‘2𝐡||β‰€πΆπ‘˜πœ”(π‘₯)‖𝑏‖Lip𝛽(πœ”)πœ”ξ€·2π‘˜+1𝐡𝛽/𝑛.(4.12) Since πœ”βˆˆπ΄1, by the HΓΆlder inequality, we get 𝒩2β‰€πΆβˆžξ“π‘˜=12(π‘˜+1)π‘›π‘’βˆ’π‘22(π‘˜βˆ’1)ξ€·2π‘˜πœ”π‘˜+1𝐡𝛽/𝑛||2π‘˜+1𝐡||πœ”(π‘₯)‖𝑏‖Lip𝛽(πœ”)ξ€œ2π‘˜+1𝐡||||𝑓(𝑧)π‘‘π‘§β‰€πΆβˆžξ“π‘˜=1π‘˜2(π‘˜+1)π‘›π‘’βˆ’π‘22(π‘˜βˆ’1)πœ”(π‘₯)‖𝑏‖Lip𝛽(πœ”)πœ”ξ€·2π‘˜+1𝐡𝛽/𝑛||2π‘˜+1𝐡||1/π‘Ÿξ‚΅ξ€œ2π‘˜+1𝐡||𝑓||(𝑧)π‘Ÿξ‚Άπ‘‘π‘§1/π‘Ÿ=πΆβˆžξ“π‘˜=1π‘˜2(π‘˜+1)π‘›π‘’βˆ’π‘22(π‘˜βˆ’1)Γ—πœ”(π‘₯)‖𝑏‖Lip𝛽(πœ”)ξƒ©πœ”(2π‘˜+1𝐡)|2π‘˜+11𝐡|πœ”ξ€·2π‘˜+1𝐡1βˆ’π›½π‘Ÿ/π‘›ξ€œ2π‘˜+1𝐡||||𝑓(𝑧)π‘Ÿξƒͺ𝑑𝑧1/π‘Ÿβ‰€πΆβˆžξ“π‘˜=1π‘˜2(π‘˜+1)π‘›π‘’βˆ’π‘22(π‘˜βˆ’1)πœ”(π‘₯)‖𝑏‖Lip𝛽(πœ”)ξ‚΅1πœ”(2π‘˜+1𝐡)1βˆ’π›½π‘Ÿ/π‘›ξ€œ2π‘˜+1𝐡||||𝑓(𝑧)π‘Ÿξ‚Άπœ”(𝑧)𝑑𝑧1/π‘Ÿβ‰€πΆβ€–π‘β€–Lip𝛽(πœ”)πœ”(π‘₯)𝑀𝛽,π‘Ÿ,πœ”π‘“(π‘₯).(4.13)
Thus, Lemma 4.6 is proved.

Lemma 4.7. Let 0<𝛿<1, 0<𝛼<1,πœ”βˆˆπ΄1, and π‘βˆˆLip𝛽(πœ”). Then for all π‘Ÿ>1 and for all π‘₯βˆˆβ„π‘›, one has 𝑀#𝐿,𝛿𝑏,πΏβˆ’π›Ό/2𝑓(π‘₯)≀𝐢‖𝑏‖Lip𝛽(πœ”)Γ—ξ€·πœ”(π‘₯)1+𝛽/𝑛𝑀𝛽,1ξ€·πΏβˆ’π›Ό/2𝑓(π‘₯)+πœ”(π‘₯)1βˆ’π›Ό/𝑛𝑀𝛼+𝛽,π‘Ÿ,πœ”π‘“(π‘₯)+πœ”(π‘₯)1+𝛽/𝑛𝑀𝛼+𝛽,1ξ€Έ.𝑓(π‘₯)(4.14)

Proof. For any given π‘₯βˆˆβ„π‘›, fix a ball 𝐡=𝐡(π‘₯0,π‘Ÿπ΅) which contains π‘₯. We decompose 𝑓=𝑓1+𝑓2, where 𝑓1=π‘“πœ’2𝐡. Observe that 𝑏,πΏβˆ’π›Ό/2𝑓=π‘βˆ’π‘2π΅ξ€ΈπΏβˆ’π›Ό/2π‘“βˆ’πΏβˆ’π›Ό/2ξ€·ξ€·π‘βˆ’π‘2𝐡𝑓1ξ€Έβˆ’πΏβˆ’π›Ό/2ξ€·ξ€·π‘βˆ’π‘2𝐡𝑓2ξ€Έπ‘’βˆ’π‘‘π΅πΏξ€·ξ€Ίπ‘,πΏβˆ’π›Ό/2𝑓=π‘’βˆ’π‘‘π΅πΏξ€Ίξ€·π‘βˆ’π‘2π΅ξ€ΈπΏβˆ’π›Ό/2π‘“βˆ’πΏβˆ’π›Ό/2ξ€·ξ€·π‘βˆ’π‘2𝐡𝑓1ξ€Έβˆ’πΏβˆ’π›Ό/2ξ€·ξ€·π‘βˆ’π‘2𝐡𝑓2.ξ€Έξ€»(4.15) Then ξ‚΅1||𝐡||ξ€œπ΅||𝑏,πΏβˆ’π›Ό/2𝑓(𝑦)βˆ’π‘’βˆ’π‘‘π΅πΏξ€·ξ€Ίπ‘,πΏβˆ’π›Ό/2𝑓||(𝑦)𝛿𝑑𝑦1/𝛿1≀𝐢||𝐡||ξ€œπ΅||𝑏(𝑦)βˆ’π‘2π΅ξ€ΈπΏβˆ’π›Ό/2𝑓||(𝑦)𝛿𝑑𝑦1/𝛿1+𝐢||𝐡||ξ€œπ΅||πΏβˆ’π›Ό/2ξ€·ξ€·π‘βˆ’π‘2𝐡𝑓1ξ€Έ(||𝑦)𝛿𝑑𝑦1/𝛿1+𝐢||𝐡||ξ€œπ΅||π‘’βˆ’π‘‘π΅πΏξ€·ξ€·π‘βˆ’π‘2π΅ξ€ΈπΏβˆ’π›Ό/2𝑓||(𝑦)𝛿𝑑𝑦1/𝛿1+𝐢||𝐡||ξ€œπ΅||π‘’βˆ’π‘‘π΅πΏξ€·πΏβˆ’π›Ό/2ξ€·ξ€·π‘βˆ’π‘2𝐡𝑓1||ξ€Έξ€Έ(𝑦)𝛿𝑑𝑦1/𝛿1+𝐢||𝐡||ξ€œπ΅||ξ€·πΏβˆ’π›Ό/2βˆ’π‘’βˆ’π‘‘π΅πΏπΏβˆ’π›Ό/2ξ€Έξ€·ξ€·π‘βˆ’π‘2𝐡𝑓2ξ€Έ||(𝑦)𝛿𝑑𝑦1/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 πœβ€²π›Ώ<1. Since πœ”βˆˆπ΄1, then it follows from HΓΆlder's inequality that ξ‚΅1𝐼≀𝐢||𝐡||ξ€œπ΅||𝑏(𝑦)βˆ’π‘2𝐡||πœπ›Ώξ‚Άπ‘‘π‘¦1/πœπ›Ώξ‚΅ξ€œπ΅||πΏβˆ’π›Ό/2||𝑓(𝑦)πœξ…žπ›Ώξ‚Άπ‘‘π‘¦1/πœξ…žπ›Ώξ‚΅1≀𝐢||𝐡||ξ€œ2𝐡||𝑏(𝑦)βˆ’π‘2𝐡||ξ€œπ‘‘π‘¦ξ‚Άξ‚΅π΅||πΏβˆ’π›Ό/2||𝑓(𝑦)𝑑𝑦≀𝐢‖𝑏‖Lip𝛽(πœ”)1||||2π΅πœ”(2𝐡)1+𝛽/π‘›ξ‚΅ξ€œπ΅||πΏβˆ’π›Ό/2||𝑓(𝑦)𝑑𝑦≀𝐢‖𝑏‖Lip𝛽(πœ”)πœ”(π‘₯)1+𝛽/𝑛𝑀𝛽,1ξ€·πΏβˆ’π›Ό/2𝑓(π‘₯).(4.17)

For 𝐼𝐼, using HΓΆlder's inequality, Kolmogorov's inequality (see page 485 [8],) and Remark 3.6, then we deduce that 1𝐼𝐼≀𝐢||𝐡||ξ€œπ΅||πΏβˆ’π›Ό/2ξ€·ξ€·π‘βˆ’π‘2𝐡𝑓1ξ€Έ(||1𝑦)𝑑𝑦≀𝐢||𝐡||||𝐡||𝛼/π‘›β€–β€–πΏβˆ’π›Ό/2ξ€·π‘βˆ’π‘2𝐡𝑓1‖‖𝐿𝑛/(π‘›βˆ’π›Ό),∞1≀𝐢||𝐡||1βˆ’π›Ό/π‘›ξ€œπ΅||𝑏(𝑦)βˆ’π‘2𝐡𝑓1||1(𝑦)𝑑𝑦≀𝐢||𝐡||1βˆ’π›Ό/π‘›ξ‚΅ξ€œ2𝐡||||𝑓(𝑦)π‘Ÿξ‚Άπ‘€(𝑦)𝑑𝑦1/π‘Ÿξ‚΅ξ€œ2𝐡||𝑏(𝑦)βˆ’π‘2𝐡||π‘Ÿξ…žπ‘€(𝑦)βˆ’π‘Ÿξ…ž/π‘Ÿξ‚Άπ‘‘π‘¦1/π‘Ÿξ…žβ‰€πΆβ€–π‘β€–Lip𝛽(πœ”)𝑀𝛼+𝛽,π‘Ÿ,πœ”ξ‚΅π‘“(π‘₯)πœ”(2𝐡)||||ξ‚Ά2𝐡1βˆ’π›Ό/𝑛≀𝐢‖𝑏‖Lip𝛽(πœ”)πœ”(π‘₯)1βˆ’π›Ό/𝑛𝑀𝛼+𝛽,π‘Ÿ,πœ”π‘“(π‘₯),(4.18) where we have used the condition that πœ”βˆˆπ΄1.

Using HΓΆlder's inequality and Lemma 4.6, we obtain that 𝐼𝐼𝐼≀𝐢‖𝑏‖Lip𝛽(πœ”)πœ”(π‘₯)𝑀𝛽,π‘Ÿ,πœ”ξ€·πΏβˆ’π›Ό/2𝑓(π‘₯).(4.19)

For 𝐼𝑉, using the estimate in II, we get 𝐢𝐼𝑉≀||𝐡||ξ€œπ΅ξ€œ2𝐡||𝑝𝑑𝐡(||||𝐿𝑦,𝑧)βˆ’π›Ό/2ξ€·ξ€·π‘βˆ’π‘2𝐡𝑓(||≀𝐢𝑧)𝑑𝑧𝑑𝑦||||ξ€œ2𝐡2𝐡||πΏβˆ’π›Ό/2ξ€·ξ€·π‘βˆ’π‘2𝐡𝑓||(𝑧)𝑑𝑧≀𝐢‖𝑏‖Lip𝛽(πœ”)πœ”(π‘₯)1βˆ’π›Ό/𝑛𝑀𝛼+𝛽,π‘Ÿ,πœ”π‘“(π‘₯).(4.20)

By virtue of Lemma 3.1, we have the following: 𝐢𝑉≀||𝐡||ξ€œπ΅ξ€œ(2𝐡)𝑐||𝐾𝛼,𝑑𝐡(||||𝑦,𝑧)𝑏(𝑧)βˆ’π‘2𝐡||𝑓(𝑧)π‘‘π‘§π‘‘π‘¦β‰€πΆβˆžξ“π‘˜=1ξ€œ2π‘˜π‘Ÿπ΅β‰€|π‘₯0βˆ’π‘§|<2π‘˜+1π‘Ÿπ΅1||π‘₯0||βˆ’π‘§π‘›βˆ’π›Όπ‘Ÿ2𝐡||π‘₯0||βˆ’π‘§2||𝑏(𝑧)βˆ’π‘2𝐡||𝑓(𝑧)π‘‘π‘§β‰€πΆβˆžξ“π‘˜=12βˆ’2π‘˜1||2π‘˜+1𝐡||1βˆ’π›Ό/π‘›ξ€œ2π‘˜+1𝐡||𝑏(𝑧)βˆ’π‘2𝐡||𝑓(𝑧)π‘‘π‘§β‰€πΆβˆžξ“π‘˜=12βˆ’2π‘˜1||2π‘˜+1𝐡||1βˆ’π›Ό/π‘›ξ€œ2π‘˜+1𝐡||𝑏(𝑧)βˆ’π‘2π‘˜+1𝐡||𝑓(𝑧)𝑑𝑧+πΆβˆžξ“π‘˜=12βˆ’2π‘˜||𝑏2π‘˜+1π΅βˆ’π‘2𝐡||1||2π‘˜+1𝐡||1βˆ’π›Ό/π‘›ξ€œ2π‘˜+1𝐡||||𝑓(𝑧)𝑑𝑧≐𝑉𝐼+𝑉𝐼𝐼.(4.21) Making use of the same argument as that of 𝐼𝐼, we have 𝑉𝐼≀𝐢‖𝑏‖Lip𝛽(πœ”)πœ”(π‘₯)1βˆ’π›Ό/𝑛𝑀𝛼+𝛽,π‘Ÿ,πœ”π‘“(π‘₯).(4.22)

Note that πœ”βˆˆπ΄1, ||𝑏2π‘˜+1π΅βˆ’π‘2𝐡||β‰€πΆπ‘˜πœ”(π‘₯)‖𝑏‖Lip𝛽(πœ”)πœ”ξ€·2π‘˜+1𝐡𝛽/𝑛.(4.23) So, the value of 𝑉𝐼𝐼 can be controlled by 𝐢‖𝑏‖Lip𝛽(πœ”)πœ”(π‘₯)1+𝛽/𝑛𝑀𝛼+𝛽,1𝑓(π‘₯).(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 (π‘Ž). 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 π‘Ÿ such that 1<π‘Ÿ<𝑝. By Lemma 4.7, we obtain the following: ‖‖𝑏,πΏβˆ’π›Ό/2ξ€»π‘“β€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”1βˆ’(1βˆ’π›Ό/𝑛)π‘ž,πœ”)‖‖𝑀≀𝐢#𝐿,𝛿𝑏,πΏβˆ’π›Ό/2ξ€»π‘“ξ€Έβ€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”1βˆ’(1βˆ’π›Ό/𝑛)π‘ž,πœ”)≀𝐢‖𝑏‖Lip𝛽(πœ”)ξ‚€β€–β€–πœ”(β‹…)1+𝛽/𝑛𝑀𝛽,1ξ€·πΏβˆ’π›Ό/2π‘“ξ€Έβ€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”1βˆ’(1βˆ’π›Ό/𝑛)π‘ž,πœ”)+β€–β€–πœ”(β‹…)1βˆ’π›Ό/𝑛𝑀𝛼+𝛽,π‘Ÿ,πœ”π‘“β€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”1βˆ’(1βˆ’π›Ό/𝑛)π‘ž,πœ”)+β€–β€–πœ”(β‹…)1+𝛽/𝑛𝑀𝛼+𝛽,1π‘“β€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”1βˆ’(1βˆ’π›Ό/𝑛)π‘ž,πœ”)≀𝐢‖𝑏‖Lip𝛽(πœ”)‖‖𝑀𝛽,1ξ€·πΏβˆ’π›Ό/2π‘“ξ€Έβ€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž/𝑝,πœ”)+‖‖𝑀𝛼+𝛽,π‘Ÿ,πœ”π‘“β€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”)+‖‖𝑀𝛼+𝛽,1π‘“β€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”π‘ž/𝑝,πœ”).(4.25) Let 1/π‘ž1=1/π‘βˆ’π›Ό/𝑛 and 1/π‘ž=1/π‘ž1βˆ’π›½/𝑛. Lemmas 4.1–4.4 yield that ‖‖𝑏,πΏβˆ’π›Ό/2ξ€»π‘“β€–β€–πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”1βˆ’(1βˆ’π›Ό/𝑛)π‘ž,πœ”)≀𝐢‖𝑏‖Lip𝛽(πœ”)ξ‚€β€–β€–πΏβˆ’π›Ό/2π‘“β€–β€–πΏπ‘ž11,π‘˜π‘ž/𝑝(πœ”π‘ž1/𝑝,πœ”)+‖𝑓‖𝐿𝑝,π‘˜(πœ”)≀𝐢‖𝑏‖Lip𝛽(πœ”)‖𝑓‖𝐿𝑝,π‘˜(πœ”).(4.26)
Now we prove (b). Let 𝐿=βˆ’Ξ” be the Laplacian on ℝ𝑛, then πΏβˆ’π›Ό/2 is the classical fractional integral 𝐼𝛼. We use the same argument as Janson [13]. Choose 𝑍0βˆˆβ„π‘› so that |𝑍0|=3. For π‘₯∈𝐡(𝑍0,2),|π‘₯|βˆ’π›Ό+𝑛 can be written as the absolutely convergent Fourier series, |π‘₯|βˆ’π›Ό+𝑛=βˆ‘π‘šβˆˆβ„€π‘›π‘Žπ‘šπ‘’π‘–βŸ¨πœˆπ‘š,π‘₯⟩ with βˆ‘π‘š|π‘Žπ‘š|<∞ since |π‘₯|βˆ’π›Ό+π‘›βˆˆπΆβˆž(𝐡(𝑍0,2)). For any π‘₯0βˆˆβ„π‘› and 𝜌>0, let 𝐡=𝐡(π‘₯0,𝜌) and 𝐡𝑍0=𝐡(π‘₯0+𝑍0𝜌,𝜌), ξ€œπ΅|||𝑏(π‘₯)βˆ’π‘π΅π‘0|||1𝑑π‘₯=||𝐡𝑍0||ξ€œπ΅|||||ξ€œπ΅π‘0|||||=1(𝑏(π‘₯)βˆ’π‘(𝑦))𝑑𝑦𝑑π‘₯πœŒπ‘›ξ€œπ΅ξƒ©ξ€œπ‘ (π‘₯)𝐡𝑍0||||(𝑏(π‘₯)βˆ’π‘(𝑦))π‘₯βˆ’π‘¦βˆ’π›Ό+𝑛||||π‘₯βˆ’π‘¦π‘›βˆ’π›Όξƒͺ𝑑𝑦𝑑π‘₯,(4.27) where βˆ«π‘ (π‘₯)=sgn(𝐡𝑍0(𝑏(π‘₯)βˆ’π‘(𝑦))𝑑𝑦). Fix π‘₯∈𝐡 and π‘¦βˆˆπ΅π‘0, then (π‘¦βˆ’π‘₯)/πœŒβˆˆπ΅π‘0,2, hence, πœŒβˆ’π›Ό+π‘›πœŒπ‘›ξ€œπ΅ξƒ©ξ€œπ‘ (π‘₯)𝐡𝑍0||||(𝑏(π‘₯)βˆ’π‘(𝑦))π‘₯βˆ’π‘¦βˆ’π›Ό+𝑛||||π‘₯βˆ’π‘¦πœŒξ‚Άπ‘›βˆ’π›Όξƒͺ𝑑𝑦𝑑π‘₯=πœŒβˆ’π›Όξ“π‘šβˆˆπ‘π‘›π‘Žπ‘šξ€œπ΅ξƒ©ξ€œπ‘ (π‘₯)𝐡𝑍0||||(𝑏(π‘₯)βˆ’π‘(𝑦))π‘₯βˆ’π‘¦π‘›βˆ’π›Όπ‘’π‘–βŸ¨πœˆπ‘š,𝑦/𝜌⟩ξƒͺπ‘’π‘‘π‘¦βˆ’π‘–βŸ¨πœˆπ‘š,π‘₯/πœŒβŸ©π‘‘π‘₯β‰€πœŒβˆ’π›Ό|||||ξ“π‘šβˆˆπ‘π‘›||π‘Žπ‘š||ξ€œπ΅ξ€Ίπ‘ (π‘₯)𝑏,πΏβˆ’π›Ό/2ξ€»ξ‚€πœ’π΅π‘0π‘’π‘–βŸ¨πœˆπ‘š,β‹…/πœŒβŸ©ξ‚(π‘₯)πœ’π΅(π‘₯)π‘’βˆ’π‘–βŸ¨πœˆπ‘š,π‘₯/𝜌⟩|||||𝑑π‘₯β‰€πœŒβˆ’π›Όξ“π‘šβˆˆπ‘π‘›||π‘Žπ‘š||‖‖𝑏,πΏβˆ’π›Ό/2ξ€»ξ‚€πœ’π΅π‘0π‘’π‘–βŸ¨πœˆπ‘š,β‹…/πœŒβŸ©ξ‚β€–β€–πΏπ‘ž,0(πœ”1βˆ’(1βˆ’π›Ό/𝑛)π‘ž,πœ”)ξ‚΅ξ€œπ΅πœ”(π‘₯)π‘žβ€²(1/π‘žβ€²βˆ’π›Ό/𝑛)𝑑π‘₯1/π‘žξ…žβ‰€πΆπœŒβˆ’π›Όξ“π‘šβˆˆπ‘π‘›||π‘Žπ‘š||β€–β€–πœ’π΅π‘0‖‖𝐿𝑝,0(πœ”)ξ‚΅ξ€œπ΅πœ”(π‘₯)π‘žβ€²(1/π‘žβ€²βˆ’π›Ό/𝑛)𝑑π‘₯1/π‘žξ…žβ‰€πΆπœ”(𝐡)1/𝑝+1/π‘žβ€²βˆ’π›Ό/𝑛=πΆπœ”(𝐡)1+𝛽/𝑛.(4.28) This implies that π‘βˆˆLip𝛽(πœ”). 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).

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