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Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 283730, 13 pages
doi:10.1155/2012/283730
Research Article

Weighted Herz Spaces and Regularity Results

Yuxing GuoΒ and Yinsheng Jiang

College of Mathematics and System Sciences, Xinjiang University, Xinjiang, Urumqi 830046, China

Received 13 April 2012; Accepted 20 July 2012

Academic Editor: Hans G.Β Feichtinger

Copyright Β© 2012 Yuxing Guo and Yinsheng Jiang. 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

It is proved that, for the nondivergence form elliptic equations βˆ‘ 𝑛 𝑖 , 𝑗 = 1 π‘Ž 𝑖 𝑗 𝑒 π‘₯ 𝑖 π‘₯ 𝑗 = 𝑓 , if 𝑓 belongs to the weighted Herz spaces 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) , then 𝑒 π‘₯ 𝑖 π‘₯ 𝑗 ∈ 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) , where 𝑒 is the π‘Š 2 , 𝑝 -solution of the equations. In order to obtain this, the authors first establish the weighted boundedness for the commutators of some singular integral operators on 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) .

1. Introduction

For a sequence πœ‘ = { πœ‘ ( π‘˜ ) } ∞ βˆ’ ∞ , πœ‘ ( π‘˜ ) > 0 , we suppose that πœ‘ satisfies doubling condition of order ( 𝑠 , 𝑑 ) and write πœ‘ ∈ 𝐷 ( 𝑠 , 𝑑 ) if there exists 𝐢 β‰₯ 1 such that 𝐢 βˆ’ 1 2 𝑠 ( π‘˜ βˆ’ 𝑗 ) ≀ πœ‘ ( π‘˜ ) πœ‘ ( 𝑗 ) ≀ 𝐢 2 𝑑 ( π‘˜ βˆ’ 𝑗 ) f o r π‘˜ > 𝑗 . ( 1 . 1 ) Let 𝐡 π‘˜ = 𝐡 ( 0 , 2 π‘˜ ) = { π‘₯ ∈ ℝ 𝑛 ∢ | π‘₯ | ≀ 2 π‘˜ } , 𝐸 π‘˜ = 𝐡 π‘˜ ⧡ 𝐡 π‘˜ βˆ’ 1 for π‘˜ ∈ β„€ , and πœ’ π‘˜ = πœ’ 𝐸 π‘˜ be the characteristic function of the set 𝐸 π‘˜ for π‘˜ ∈ β„€ . Suppose that 𝑀 is a weight function on ℝ 𝑛 . For 1 < 𝑝 < ∞ , 0 < π‘ž < ∞ , the weighted Herz space is defined by 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( ℝ 𝑛  ) = 𝑓 ∢ 𝑓 i s a m e a s u r a b l e f u n c t i o n o n ℝ 𝑛 , β€– 𝑓 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 )  , < ∞ ( 1 . 2 ) where β€– 𝑓 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) =  ∞  π‘˜ = βˆ’ ∞ πœ‘ ( π‘˜ ) π‘ž β€– β€– 𝑓 πœ’ π‘˜ β€– β€– π‘ž 𝐿 𝑝 ( 𝑀 ) ξƒͺ 1 / π‘ž , β€– 𝑓 β€– 𝐿 𝑝 ( 𝑀 ) = ξ‚΅ ξ€œ ℝ 𝑛 | | | | 𝑓 ( π‘₯ ) 𝑝 ξ‚Ά 𝑀 ( π‘₯ ) 𝑑 π‘₯ 1 / 𝑝 . ( 1 . 3 )

Beurling in [1] introduced the Beurling algebras, and Herz in [2] generalized these spaces; many studies have been done for Herz spaces (see, e.g, [3, 4]). Weighted Herz spaces are also considered in [5, 6]. Lu and Tao in [7] studied nondivergence form elliptic equations on Morrey-Herz spaces, which are more general spaces. Ragusa in [8, 9] obtained some regularity results to the divergence form elliptic and parabolic equations on homogeneous Herz spaces.

The paper is organized as follows. In Section 2, we give some basic notions. In this section, we recall also continuity results regarding the Calderón-Zygmund singular integral operators that will appear in the representation formula of the 𝑒 π‘₯ 𝑖 π‘₯ 𝑗 estimates. In Section 3, we prove the boundedness of the commutators of some singular integral operators on weighted Herz spaces. In Section 4, we study the interior estimates on weighted Herz spaces for the solutions of some nondivergence elliptic equations βˆ‘ 𝑛 𝑖 , 𝑗 = 1 π‘Ž 𝑖 𝑗 𝑒 π‘₯ 𝑖 π‘₯ 𝑗 = 𝑓 , and we prove that if 𝑓 ∈ 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) , then 𝑒 π‘₯ 𝑖 π‘₯ 𝑗 ∈ 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) , where 𝑒 is the π‘Š 2 , 𝑝 -solution of the equations.

Throughout this paper, unless otherwise indicated, 𝐢 will be used to denote a positive constant that is not necessarily the same at each occurrence.

2. Preliminaries

We begin this section with some properties of 𝐴 𝑝 weights classes which play important role in the proofs of our main results. For more about 𝐴 𝑝 classes, we can refer to [10, 11].

Definition 2.1 ( 𝐴 𝑝 w e i g h t s ( 1 ≀ 𝑝 < ∞ ) ). Let 𝑀 ( π‘₯ ) β‰₯ 0 and 𝑀 ( π‘₯ ) ∈ 𝐿 1 l o c ( ℝ 𝑛 ) . One says that 𝑀 ∈ 𝐴 𝑝 for 1 < 𝑝 < ∞ if there exists a constant 𝐢 such that for every ball 𝐡 βŠ‚ ℝ 𝑛 , s u p 𝐡 ξ‚» 1 | | 𝐡 | | ξ€œ 𝐡 1 𝑀 ( π‘₯ ) 𝑑 π‘₯ ξ‚Ό ξ‚» | | 𝐡 | | ξ€œ 𝐡 𝑀 ( π‘₯ ) 1 βˆ’ 𝑝 β€² ξ‚Ό 𝑑 π‘₯ 𝑝 βˆ’ 1 ≀ 𝐢 ( 2 . 1 ) holds, here and below, 1 / 𝑝 + 1 / 𝑝 β€² = 1 . One says that 𝑀 ∈ 𝐴 1 if there exists a positive constant 𝐢 such that 1 | | 𝐡 | | ξ€œ 𝐡 𝑀 ( π‘₯ ) 𝑑 π‘₯ ≀ 𝐢 e s s i n f π‘₯ ∈ 𝐡 𝑀 ( π‘₯ ) . ( 2 . 2 ) The smallest constant appearing in (2.1) or (2.2) is called the 𝐴 𝑝 constant of 𝑀 , denoted by 𝐢 𝑀 .

Lemma 2.2. Let 1 ≀ 𝑝 < ∞ and 𝑀 ∈ 𝐴 𝑝 . Then the following statements are true:(1)(strong doubling) there exists a constant 𝐢 such that 𝑀 ξ€· 𝐡 π‘˜ ξ€Έ 𝑀 ξ€· 𝐡 𝑗 ξ€Έ ≀ 𝐢 2 𝑛 𝑝 ( π‘˜ βˆ’ 𝑗 ) f o r π‘˜ > 𝑗 , ( 2 . 3 ) (2)(centered reverse doubling) for some 𝛿 > 0 , 𝑀 ∈ 𝑅 𝐷 ( 𝛿 ) , that is, 𝑀 ξ€· 𝐡 π‘˜ ξ€Έ 𝑀 ξ€· 𝐡 𝑗 ξ€Έ β‰₯ 𝐢 2 𝛿 ( π‘˜ βˆ’ 𝑗 ) f o r π‘˜ > 𝑗 , ( 2 . 4 ) (3)for 1 < 𝑝 < ∞ , one has 𝑀 ∈ 𝐴 𝑝    for some 𝑝 < 𝑝 ,(4)there exist two constants 𝐢 and 𝛿 > 0 such that for any measurable set 𝐡 βŠ‚ 𝐸 , 𝑀 ( 𝐡 ) ξ‚΅ | | 𝐡 | | 𝑀 ( 𝐸 ) ≀ 𝐢 | | 𝐸 | | ξ‚Ά 𝛿 . ( 2 . 5 ) If 𝑀 satisfies (2.5), one says 𝑀 ∈ 𝐴 ∞ . Obviously, 𝐴 ∞ = ⋃ 1 ≀ 𝑝 < ∞ 𝐴 𝑝 ,(5)for all ( 1 / 𝑝 ) + ( 1 / 𝑝 β€² ) = 1 , one has 𝑀 1 βˆ’ 𝑝 ξ…ž ∈ 𝐴 𝑝 ξ…ž .

Remark 2.3. Note that ∫ 𝑀 ( 𝐸 ) = 𝐸 𝑀 ( π‘₯ ) 𝑑 π‘₯ and 𝑀 𝑝 ( 𝐸 ) 1 / 𝑝 ∫ = ( 𝐸 𝑀 𝑝 ( π‘₯ ) 𝑑 π‘₯ ) 1 / 𝑝 .

Definition 2.4. Let Ξ© βŠ‚ ℝ 𝑛 be an open set. One says that any 𝑓 ∈ 𝐿 1 l o c ( Ξ© ) is in the bounded mean oscillation spaces B M O ( Ξ© ) if s u p 𝛾 > 0 , π‘₯ ∈ 𝐡 𝛾 ( π‘₯ ) βŠ‚ Ξ© 1 | | 𝐡 𝛾 ( | | ξ€œ π‘₯ ) 𝐡 𝛾 ( π‘₯ ) | | | 𝑓 ( 𝑦 ) βˆ’ 𝑓 𝐡 𝛾 ( π‘₯ ) | | | 𝑑 𝑦 ≑ β€– 𝑓 β€– βˆ— < ∞ , ( 2 . 6 ) where 𝑓 𝐡 𝛾 ( π‘₯ ) is the average over 𝐡 𝛾 ( π‘₯ ) of 𝑓 . Moreover, for any 𝑓 ∈ B M O ( Ξ© ) and π‘Ÿ > 0 , one sets s u p 𝛾 ≀ π‘Ÿ , π‘₯ ∈ 𝐡 𝛾 ( π‘₯ ) βŠ‚ Ξ© 1 | | 𝐡 𝛾 ( | | ξ€œ π‘₯ ) 𝐡 𝛾 ( π‘₯ ) | | | 𝑓 ( 𝑦 ) βˆ’ 𝑓 𝐡 𝛾 ( π‘₯ ) | | | 𝑑 𝑦 ≑ πœ‚ ( π‘Ÿ ) . ( 2 . 7 ) One says that any 𝑓 ∈ B M O ( Ξ© ) is in the vanishing mean oscillation spaces V M O ( Ξ© ) if πœ‚ ( π‘Ÿ ) β†’ 0 as π‘Ÿ β†’ 0 and refer to πœ‚ ( π‘Ÿ ) as the modulus of 𝑓 .

Remark 2.5. 𝑓 ∈ B M O ( ℝ 𝑛 ) or V M O ( ℝ 𝑛 ) if 𝐡 ranges in the class of balls of ℝ 𝑛 .

Lemma 2.6 (see [12, Theorem 5]). Let 𝑀 ∈ 𝐴 ∞ . Then the norm of B M O ( 𝑀 ) is equivalent to the norm of B M O ( ℝ 𝑛 ) , where ξ‚» B M O ( 𝑀 ) = π‘Ž ∢ β€– π‘Ž β€– βˆ— , 𝑀 1 = s u p ξ€œ 𝑀 ( 𝐡 ) 𝐡 | | π‘Ž ( π‘₯ ) βˆ’ π‘Ž 𝐡 , 𝑀 | | ξ‚Ό , π‘Ž 𝑀 ( π‘₯ ) 𝑑 π‘₯ 𝐡 , 𝑀 = 1 ξ€œ 𝑀 ( 𝐡 ) 𝐡 π‘Ž ( 𝑧 ) 𝑀 ( 𝑧 ) 𝑑 𝑧 . ( 2 . 8 )

Definition 2.7. Let 𝐾 ∢ ℝ 𝑛 ⧡ { 0 } β†’ ℝ . One says that 𝐾 ( π‘₯ ) is a constant Calderón-Zygmund kernel ( constant 𝐢 - 𝑍 kernel ) if(i) 𝐾 ∈ 𝐢 ∞ ( ℝ 𝑛 ⧡ { 0 } ) ,(ii) 𝐾 is homogeneous of degree βˆ’ 𝑛 ,(iii) ∫ 𝑆 𝑛 βˆ’ 1 𝐾 ( π‘₯ ) 𝑑 𝜎 = 0 , 𝑆 𝑛 βˆ’ 1 = { π‘₯ ∈ ℝ 𝑛 ∢ | π‘₯ | = 1 } .

Definition 2.8. Let Ξ© be an open set of ℝ 𝑛 and 𝐾 ∢ Ξ© Γ— { ℝ 𝑛 ⧡ { 0 } } β†’ ℝ . One says that 𝐾 ( π‘₯ , 𝑦 ) is a variable Calderón-Zygmund kernel ( variable 𝐢 - 𝑍 kernel ) on Ξ© if(i) 𝐾 ( π‘₯ , β‹… ) is a constant 𝐢 - 𝑍 kernel for a.e. π‘₯ ∈ Ξ© ,(ii) m a x | 𝑗 | ≀ 2 𝑛 β€– ( πœ• 𝑗 / πœ• 𝑧 𝑗 ) 𝐾 ( π‘₯ , 𝑧 ) β€– 𝐿 ∞ ( Ξ© Γ— 𝑆 𝑛 βˆ’ 1 ) ≑ 𝑀 < ∞ .
Let 𝐾 be a constant or a variable 𝐢 - 𝑍 kernel on Ξ© . One defines the corresponding 𝐢 - 𝑍 operator by ξ€œ 𝑇 𝑓 ( π‘₯ ) = P . V . ℝ 𝑛 ξ€œ 𝐾 ( π‘₯ βˆ’ 𝑦 ) 𝑓 ( 𝑦 ) 𝑑 𝑦 o r 𝑇 𝑓 ( π‘₯ ) = P . V . Ξ© 𝐾 ( π‘₯ , π‘₯ βˆ’ 𝑦 ) 𝑓 ( 𝑦 ) 𝑑 𝑦 . ( 2 . 9 )

Lemma 2.9 (see [5, Theorem 3]). Let 1 < 𝑝 < ∞ , 0 < π‘ž < ∞ , 𝛿 > 0 . One assumes that(i) πœ‘ ∈ 𝐷 ( 𝑠 , 𝑑 ) , where βˆ’ ( 𝛿 / 𝑝 ) < 𝑠 ≀ 𝑑 < 𝑛 ( 1 βˆ’ ( 1 / 𝑝 ) ) ,(ii) 𝑀 ∈ 𝐴 π‘Ÿ , where π‘Ÿ = m i n ( 𝑝 , 𝑝 ( 1 βˆ’ ( 𝑑 / 𝑛 ) ) ) ,(iii) 𝑀 ∈ 𝑅 𝐷 ( 𝛿 ) .
If 𝐾 is a constant or a variable 𝐢 - 𝑍 kernel on ℝ 𝑛 and 𝑇 is the corresponding 𝐢 - 𝑍 operator, then there exists a constant 𝐢 such that for all 𝑓 ∈ 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( ℝ 𝑛 ) , β€– 𝑇 𝑓 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( ℝ 𝑛 ) ≀ 𝐢 β€– 𝑓 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( ℝ 𝑛 ) . ( 2 . 1 0 )

From this lemma, by a proof similar to that of Theorem  2.11 in [13], we obtain the following corollary.

Corollary 2.10. Let 1 < 𝑝 < ∞ , 0 < π‘ž < ∞ , 𝛿 > 0 , and Ξ© be an open set of ℝ n . One assumes that(i) πœ‘ ∈ 𝐷 ( 𝑠 , 𝑑 ) , where βˆ’ ( 𝛿 / 𝑝 ) < 𝑠 ≀ 𝑑 < 𝑛 ( 1 βˆ’ ( 1 / 𝑝 ) ) ,(ii) 𝑀 ∈ 𝐴 π‘Ÿ , where π‘Ÿ = m i n ( 𝑝 , 𝑝 ( 1 βˆ’ ( 𝑑 / 𝑛 ) ) ) ,(iii) 𝑀 ∈ 𝑅 𝐷 ( 𝛿 ) .If 𝐾 is a constant or a variable 𝐢 - 𝑍 kernel on Ξ© , and 𝑇 is the corresponding 𝐢 - 𝑍 operator, then there exists a constant 𝐢 such that for all 𝑓 ∈ 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( Ξ© ) , β€– 𝑇 𝑓 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( Ξ© ) ≀ 𝐢 β€– 𝑓 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( Ξ© ) . ( 2 . 1 1 )

3. Weighted Boundedness of Commutators

The aim of this section is to set up the weighted boundedness for the commutators formed by 𝑇 and B M O ( ℝ 𝑛 ) functions, where [ π‘Ž , 𝑇 ] 𝑓 ( π‘₯ ) = 𝑇 ( π‘Ž 𝑓 ) ( π‘₯ ) βˆ’ π‘Ž ( π‘₯ ) 𝑇 ( 𝑓 ) ( π‘₯ ) . This kind of operators is useful in lots of different fields, see, for example, [13] as well as [14], then we consider important in themselves the related below results.

Lemma 3.1 (see [10, Theorem 7 . 1 . 6 ]). Let π‘Ž ∈ B M O ( ℝ 𝑛 ) . Then for any ball 𝐡 βŠ‚ ℝ 𝑛 , there exist constants 𝐢 1 , 𝐢 2 such that for all 𝛼 > 0 , | | ξ€½ | | π‘₯ ∈ 𝐡 ∢ π‘Ž ( π‘₯ ) βˆ’ π‘Ž 𝐡 | | ξ€Ύ | | > 𝛼 ≀ 𝐢 1 | | 𝐡 | | 𝑒 βˆ’ 𝐢 2 𝛼 / β€– π‘Ž β€– βˆ— . ( 3 . 1 ) The inequality (3.1) is also called John-Nirenberg inequality.

Theorem 3.2. Let 1 < 𝑝 < ∞ , 0 < π‘ž < ∞ , 𝛿 > 0 , and π‘Ž ∈ B M O ( ℝ 𝑛 ) . One assumes that(i) πœ‘ ∈ 𝐷 ( 𝑠 , 𝑑 ) , where βˆ’ ( 𝛿 / 𝑝 ) < 𝑠 ≀ 𝑑 < 𝑛 ( 1 βˆ’ ( 1 / 𝑝 ) ) ,(ii) 𝑀 ∈ 𝐴 π‘Ÿ , where π‘Ÿ = m i n ( 𝑝 , 𝑝 ( 1 βˆ’ ( 𝑑 / 𝑛 ) ) ) ,(iii) 𝑀 ∈ 𝑅 𝐷 ( 𝛿 ) .If a linear operator 𝑇 satisfies | | | | ξ€œ 𝑇 ( 𝑓 ) ( π‘₯ ) ≀ 𝐢 ℝ 𝑛 | | | | 𝑓 ( 𝑦 ) | | | | π‘₯ βˆ’ 𝑦 𝑛 𝑑 𝑦 , π‘₯ βˆ‰ s u p p 𝑓 , ( 3 . 2 ) for any 𝑓 ∈ 𝐿 1 l o c ( ℝ 𝑛 ) and [ π‘Ž , 𝑇 ] is bounded on 𝐿 𝑝 ( 𝑀 ) , then [ π‘Ž , 𝑇 ] is also bounded on 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) .

Proof. Let 𝑓 ∈ 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( ℝ 𝑛 ) and π‘Ž ∈ B M O ( ℝ 𝑛 ) , we write 𝑓 ( π‘₯ ) = ∞  𝑗 = βˆ’ ∞ 𝑓 ( π‘₯ ) πœ’ 𝑗 ( π‘₯ ) = ∞  𝑗 = βˆ’ ∞ 𝑓 𝑗 ( π‘₯ ) . ( 3 . 3 ) Then, we have β€– [ ] π‘Ž , 𝑇 𝑓 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) βŽ› ⎜ ⎜ ⎝ ≀ 𝐢 ∞  π‘˜ = βˆ’ ∞ πœ‘ ( π‘˜ ) π‘ž  π‘˜ βˆ’ 2  𝑗 = βˆ’ ∞ β€– β€– ξ€· [ ] 𝑓 π‘Ž , 𝑇 𝑗 ξ€Έ πœ’ π‘˜ β€– β€– 𝐿 𝑝 ( 𝑀 ) ξƒͺ π‘ž ⎞ ⎟ ⎟ ⎠ 1 / π‘ž βŽ› ⎜ ⎜ ⎝ + 𝐢 ∞  π‘˜ = βˆ’ ∞ πœ‘ ( π‘˜ ) π‘ž  π‘˜ + 1  𝑗 = π‘˜ βˆ’ 1 β€– β€– ξ€· [ ] 𝑓 π‘Ž , 𝑇 𝑗 ξ€Έ πœ’ π‘˜ β€– β€– 𝐿 𝑝 ( 𝑀 ) ξƒͺ π‘ž ⎞ ⎟ ⎟ ⎠ 1 / π‘ž  + 𝐢 ∞  π‘˜ = βˆ’ ∞ πœ‘ ( π‘˜ ) π‘ž  ∞  𝑗 = π‘˜ + 2 β€– β€– ξ€· [ ] 𝑓 π‘Ž , 𝑇 𝑗 ξ€Έ πœ’ π‘˜ β€– β€– 𝐿 𝑝 ( 𝑀 ) ξƒͺ π‘ž ξƒͺ 1 / π‘ž = I + I I + I I I . ( 3 . 4 ) For II, by the 𝐿 𝑝 ( 𝑀 ) boundedness of [ π‘Ž , 𝑇 ] , we have βŽ› ⎜ ⎜ ⎝ I I ≀ 𝐢 ∞  π‘˜ = βˆ’ ∞ πœ‘ ( π‘˜ ) π‘ž  π‘˜ + 1  𝑗 = π‘˜ βˆ’ 1 β€– π‘Ž β€– π‘ž βˆ— β€– β€– 𝑓 𝑗 πœ’ π‘˜ β€– β€– 𝐿 𝑝 ( 𝑀 ) ξƒͺ π‘ž ⎞ ⎟ ⎟ ⎠ 1 / π‘ž ≀ 𝐢 β€– π‘Ž β€– βˆ— β€– 𝑓 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) . ( 3 . 5 ) For I , note that when π‘₯ ∈ 𝐸 π‘˜ , 𝑦 ∈ 𝐸 𝑗 , and 𝑗 ≀ π‘˜ βˆ’ 2 , | π‘₯ βˆ’ 𝑦 | ∼ | π‘₯ | . So from the condition (3.2), we have | | [ ] 𝑓 π‘Ž , 𝑇 𝑗 | | ξ€œ ≀ 𝐢 ℝ 𝑛 | | | | π‘Ž ( π‘₯ ) βˆ’ π‘Ž ( 𝑦 ) | | | | π‘₯ βˆ’ 𝑦 𝑛 | | 𝑓 𝑗 | | ( 𝑦 ) 𝑑 𝑦 ≀ 𝐢 2 βˆ’ 𝑛 π‘˜ | | π‘Ž ( π‘₯ ) βˆ’ π‘Ž 𝐡 π‘˜ , 𝑀 | | ξ€œ ℝ 𝑛 | | 𝑓 𝑗 | | ( 𝑦 ) 𝑑 𝑦 + 𝐢 2 βˆ’ 𝑛 π‘˜ | | | π‘Ž 𝐡 π‘˜ , 𝑀 βˆ’ π‘Ž 𝐡 𝑗 , 𝑀 | | | ξ€œ ℝ 𝑛 | | 𝑓 𝑗 | | ( 𝑦 ) 𝑑 𝑦 + 𝐢 2 βˆ’ 𝑛 π‘˜ ξ€œ ℝ 𝑛 | | | π‘Ž ( 𝑦 ) βˆ’ π‘Ž 𝐡 𝑗 , 𝑀 β€– β€– 𝑓 𝑗 ( | | | 𝑦 ) 𝑑 𝑦 . ( 3 . 6 ) Thus, β€– β€– ξ€· [ ] 𝑓 π‘Ž , 𝑇 𝑗 ξ€Έ πœ’ π‘˜ β€– β€– 𝐿 𝑝 ( 𝑀 ) ≀ 𝐢 2 βˆ’ 𝑛 π‘˜ β€– β€– ξ€· π‘Ž ( π‘₯ ) βˆ’ π‘Ž 𝐡 π‘˜ , 𝑀 ξ€Έ πœ’ π‘˜ β€– β€– 𝐿 𝑝 ( 𝑀 ) ξ€œ ℝ 𝑛 | | 𝑓 𝑗 ( | | 𝑦 ) 𝑑 𝑦 + 𝐢 2 βˆ’ 𝑛 π‘˜ | | | π‘Ž 𝐡 π‘˜ , 𝑀 βˆ’ π‘Ž 𝐡 𝑗 , 𝑀 | | | 𝑀 ξ€· 𝐡 π‘˜ ξ€Έ 1 / 𝑝 ξ€œ ℝ 𝑛 | | 𝑓 𝑗 | | ( 𝑦 ) 𝑑 𝑦 + 𝐢 2 βˆ’ 𝑛 π‘˜ 𝑀 ξ€· 𝐡 π‘˜ ξ€Έ 1 / 𝑝 ξ€œ ℝ 𝑛 | | | π‘Ž ( 𝑦 ) βˆ’ π‘Ž 𝐡 𝑗 , 𝑀 β€– β€– 𝑓 𝑗 | | | = 𝐽 ( 𝑦 ) 𝑑 𝑦 1 + 𝐽 2 + 𝐽 3 . ( 3 . 7 ) According to Lemma 2.2, 𝑀 ∈ 𝐴 π‘Ÿ for some π‘Ÿ < π‘Ÿ . By Hölder’s inequality and Lemma 2.6, 𝐽 1 ≀ 𝐢 2 βˆ’ 𝑛 π‘˜ β€– π‘Ž β€– βˆ— 𝑀 ξ€· 𝐡 π‘˜ ξ€Έ 1 / 𝑝 β€– β€– 𝑓 𝑗 β€– β€– 𝐿 𝑝 ( 𝑀 ) 𝑀 βˆ’ 𝑝 β€² / 𝑝 ξ€· 𝐡 𝑗 ξ€Έ 1 / 𝑝 β€² = 𝐢 2 βˆ’ 𝑛 π‘˜ β€– π‘Ž β€– βˆ— β€– β€– 𝑓 𝑗 β€– β€– 𝐿 𝑝 ( 𝑀 ) 𝑀 βˆ’ 𝑝 β€² / 𝑝 ξ€· 𝐡 𝑗 ξ€Έ 1 / 𝑝 β€² 𝑀 ξ€· 𝐡 𝑗 ξ€Έ 1 / 𝑝  𝑀 ξ€· 𝐡 π‘˜ ξ€Έ 𝑀 ξ€· 𝐡 𝑗 ξ€Έ ξƒͺ 1 / 𝑝 ≀ 𝐢 2 βˆ’ 𝑛 π‘˜ β€– π‘Ž β€– βˆ— β€– β€– 𝑓 𝑗 β€– β€– 𝐿 𝑝 ( 𝑀 ) | | 𝐡 𝑗 | | 2 𝑛 π‘Ÿ ( π‘˜ βˆ’ 𝑗 ) / 𝑝 ≀ 𝐢 2 π‘˜ ( βˆ’ 𝑛 + ( π‘Ÿ 𝑛 / 𝑝 ) ) 2 𝑛 𝑗 ( 1 βˆ’ ( π‘Ÿ / 𝑝 ) ) β€– π‘Ž β€– βˆ— β€– β€– 𝑓 𝑗 β€– β€– 𝐿 𝑝 ( 𝑀 ) . ( 3 . 8 ) It is easy to see that | π‘Ž 𝐡 π‘˜ , 𝑀 βˆ’ π‘Ž 𝐡 𝑗 , 𝑀 | ≀ 𝐢 ( π‘˜ βˆ’ 𝑗 ) β€– π‘Ž β€– βˆ— . Therefore, similarly to 𝐽 1 , we have 𝐽 2 ≀ 𝐢 ( π‘˜ βˆ’ 𝑗 ) 2 βˆ’ 𝑛 π‘˜ β€– π‘Ž β€– βˆ— 𝑀 ξ€· 𝐡 π‘˜ ξ€Έ 1 / 𝑝 β€– β€– 𝑓 𝑗 β€– β€– 𝐿 𝑝 ( 𝑀 ) 𝑀 βˆ’ 𝑝 β€² / 𝑝 ξ€· 𝐡 𝑗 ξ€Έ 1 / 𝑝 β€² ≀ 𝐢 π‘˜ 2 π‘˜ ( βˆ’ 𝑛 + ( π‘Ÿ 𝑛 / 𝑝 ) ) 𝑗 2 𝑛 𝑗 ( 1 βˆ’ ( π‘Ÿ / 𝑝 ) ) β€– π‘Ž β€– βˆ— β€– β€– 𝑓 𝑗 β€– β€– 𝐿 𝑝 ( 𝑀 ) . ( 3 . 9 ) Now, we establish the estimate for term 𝐽 3 , 𝐽 3 ≀ 𝐢 2 βˆ’ 𝑛 π‘˜ 𝑀 ξ€· 𝐡 π‘˜ ξ€Έ 1 / 𝑝 β€– β€– 𝑓 𝑗 β€– β€– 𝐿 𝑝 ( 𝑀 )  ξ€œ 𝐡 𝑗 | | | π‘Ž ( 𝑦 ) βˆ’ π‘Ž 𝐡 𝑗 , 𝑀 | | | 𝑝 β€² 𝑀 1 βˆ’ 𝑝 β€² ξƒͺ ( 𝑦 ) 𝑑 𝑦 1 / 𝑝 β€² . ( 3 . 1 0 ) For the simplicity of analysis, we denote 𝐻 as  ξ€œ 𝐡 𝑗 | | | π‘Ž ( 𝑦 ) βˆ’ π‘Ž 𝐡 𝑗 , 𝑀 | | | 𝑝 β€² 𝑀 1 βˆ’ 𝑝 β€² ξƒͺ ( 𝑦 ) 𝑑 𝑦 1 / 𝑝 β€² . ( 3 . 1 1 ) By an elementary estimate, we have  ξ€œ 𝐻 ≀ 𝐢 𝐡 𝑗  | | | π‘Ž ( 𝑦 ) βˆ’ π‘Ž 𝐡 𝑗 , 𝑀 β€² 1 βˆ’ 𝑝 | | | + | | | π‘Ž 𝐡 𝑗 , 𝑀 β€² 1 βˆ’ 𝑝 βˆ’ π‘Ž 𝐡 𝑗 , 𝑀 | | | ξ‚„ 𝑝 β€² 𝑀 1 βˆ’ 𝑝 β€² ξƒͺ ( 𝑦 ) 𝑑 𝑦 1 / 𝑝 β€² ≀ 𝐢 β€– π‘Ž β€– B M O ( 𝑀 β€² 1 βˆ’ 𝑝 ) 𝑀 1 βˆ’ 𝑝 β€² ξ€· 𝐡 𝑗 ξ€Έ 1 / 𝑝 β€² + | | | π‘Ž 𝐡 𝑗 , 𝑀 β€² 1 βˆ’ 𝑝 βˆ’ π‘Ž 𝐡 𝑗 , 𝑀 | | | 𝑀 1 βˆ’ 𝑝 β€² ξ€· 𝐡 𝑗 ξ€Έ 1 / 𝑝 β€² . ( 3 . 1 2 ) Note that | | | π‘Ž 𝐡 𝑗 , 𝑀 β€² 1 βˆ’ 𝑝 βˆ’ π‘Ž 𝐡 𝑗 , 𝑀 | | | ≀ | | | π‘Ž 𝐡 𝑗 , 𝑀 β€² 1 βˆ’ 𝑝 βˆ’ π‘Ž 𝐡 𝑗 | | | + | | | π‘Ž 𝐡 𝑗 βˆ’ π‘Ž 𝐡 𝑗 , 𝑀 | | | = 𝐽 3 1 + 𝐽 3 2 . ( 3 . 1 3 ) Combining (2.5) with (3.1), 𝐽 3 2 = 1 𝑀 ξ€· 𝐡 𝑗 ξ€Έ ξ€œ 𝐡 𝑗 | | | π‘Ž ( 𝑦 ) βˆ’ π‘Ž 𝐡 𝑗 | | | = 1 𝑀 ( 𝑦 ) 𝑑 𝑦 𝑀 ξ€· 𝐡 𝑗 ξ€Έ ξ€œ ∞ 0 𝑀 ξ‚€  π‘₯ ∈ 𝐡 𝑗 ∢ | | | π‘Ž ( 𝑦 ) βˆ’ π‘Ž 𝐡 𝑗 | | | ξ€œ > 𝛼   𝑑 𝛼 ≀ 𝐢 ∞ 0 𝑒 βˆ’ 𝐢 2 𝛼 𝛿 / β€– π‘Ž β€– βˆ— 𝑑 𝛼 ≀ 𝐢 . ( 3 . 1 4 ) In the same manner, we can see that 𝐽 3 1 ≀ 𝐢 . ( 3 . 1 5 ) By Lemma 2.6, we get 𝐽 3 ≀ 𝐢 2 βˆ’ 𝑛 π‘˜ β€– π‘Ž β€– βˆ— 𝑀 ξ€· 𝐡 π‘˜ ξ€Έ 1 / 𝑝 β€– β€– 𝑓 𝑗 β€– β€– 𝐿 𝑝 ( 𝑀 ) 𝑀 βˆ’ 𝑝 β€² / 𝑝 ξ€· 𝐡 𝑗 ξ€Έ 1 / 𝑝 β€² ≀ 𝐢 2 π‘˜ ( βˆ’ 𝑛 + ( π‘Ÿ 𝑛 / 𝑝 ) ) 2 𝑛 𝑗 ( 1 βˆ’ ( π‘Ÿ / 𝑝 ) ) β€– π‘Ž β€– βˆ— β€– β€– 𝑓 𝑗 β€– β€– 𝐿 𝑝 ( 𝑀 ) . ( 3 . 1 6 ) Using hypotheses πœ‘ ∈ 𝐷 ( s , 𝑑 ) and the estimates of 𝐽 1 , 𝐽 2 , and 𝐽 3 , we obtain the following inequality: I ≀ 𝐢 β€– π‘Ž β€– βˆ—   π‘˜ 2 π‘˜ ( βˆ’ 𝑛 + ( π‘Ÿ 𝑛 / 𝑝 ) + 𝑑 ) π‘ž β‹…   𝑗 ≀ π‘˜ βˆ’ 2 2 𝑛 𝑗 ( 1 βˆ’ ( π‘Ÿ / 𝑝 ) βˆ’ ( 𝑑 / 𝑛 ) ) β€– β€– πœ‘ ( 𝑗 ) 𝑓 πœ’ 𝑗 β€– β€– 𝐿 𝑝 ( 𝑀 ) ξƒͺ π‘ž ξƒͺ 1 / π‘ž + 𝐢 β€– π‘Ž β€– βˆ—   π‘˜ ξ‚€ π‘˜ 2 π‘˜ ( βˆ’ 𝑛 + ( π‘Ÿ 𝑛 / 𝑝 ) + 𝑑 )  π‘ž β‹…   𝑗 ≀ π‘˜ βˆ’ 2 𝑗 2 𝑛 𝑗 ( 1 βˆ’ ( π‘Ÿ / 𝑝 ) βˆ’ ( 𝑑 / 𝑛 ) ) β€– β€– πœ‘ ( 𝑗 ) 𝑓 πœ’ 𝑗 β€– β€– 𝐿 𝑝 ( 𝑀 ) ξƒͺ π‘ž ξƒͺ 1 / π‘ž = I 1 + I 2 . ( 3 . 1 7 ) When π‘ž ≀ 1 , we have I 1 ≀ 𝐢 β€– π‘Ž β€– βˆ—   𝑗 2 𝑛 𝑗 ( 1 βˆ’ ( π‘Ÿ / 𝑝 ) βˆ’ ( 𝑑 / 𝑛 ) ) π‘ž πœ‘ ( 𝑗 ) π‘ž β€– β€– 𝑓 πœ’ 𝑗 β€– β€– π‘ž 𝐿 𝑝 ( 𝑀 ) β‹… ∞  π‘˜ = 𝑗 + 2 2 π‘˜ ( βˆ’ 𝑛 + ( π‘Ÿ 𝑛 / 𝑝 ) + 𝑑 ) π‘ž ξƒͺ 1 / π‘ž ≀ 𝐢 β€– π‘Ž β€– βˆ— β€– 𝑓 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) , ( 3 . 1 8 ) because βˆ’ 𝑛 + 𝑛 π‘Ÿ / 𝑝 + 𝑑 ≀ 0 , that is, βˆ’ 𝑛 + 𝑛 π‘Ÿ / 𝑝 + 𝑑 < 0 .
When π‘ž > 1 , we take πœ€ > 0 such that βˆ’ 𝑛 + 𝑛 π‘Ÿ / 𝑝 + 𝑑 + 𝑛 πœ€ < 0 . Then I 1 ≀ 𝐢 β€– π‘Ž β€– βˆ— ⎑ ⎒ ⎒ ⎣  π‘˜ 2 π‘˜ ( βˆ’ 𝑛 + ( π‘Ÿ 𝑛 / 𝑝 ) + 𝑑 ) π‘ž β‹… βŽ› ⎜ ⎜ ⎝ π‘˜ βˆ’ 2  𝑗 = βˆ’ ∞ 2 𝑛 𝑗 ( 1 βˆ’ ( π‘Ÿ / 𝑝 ) βˆ’ ( 𝑑 / 𝑛 ) βˆ’ πœ€ ) π‘ž πœ‘ ( 𝑗 ) π‘ž β€– β€– 𝑓 πœ’ 𝑗 β€– β€– π‘ž 𝐿 𝑝 ( 𝑀 ) β‹…  π‘˜ βˆ’ 2  𝑗 = βˆ’ ∞ 2 𝑛 πœ€ π‘ž β€² 𝑗 ξƒͺ π‘ž / π‘ž β€² ⎞ ⎟ ⎟ ⎠ ⎀ βŽ₯ βŽ₯ ⎦ 1 / π‘ž ≀ 𝐢 β€– π‘Ž β€– βˆ—   π‘˜ 2 π‘˜ ( βˆ’ 𝑛 + ( π‘Ÿ 𝑛 / 𝑝 ) + 𝑑 + 𝑛 πœ€ ) π‘ž β‹…  π‘˜ βˆ’ 2  𝑗 = βˆ’ ∞ 2 𝑛 𝑗 ( 1 βˆ’ ( π‘Ÿ / 𝑝 ) βˆ’ ( 𝑑 / 𝑛 ) βˆ’ πœ€ ) π‘ž πœ‘ ( 𝑗 ) π‘ž β€– β€– 𝑓 πœ’ 𝑗 β€– β€– π‘ž 𝐿 𝑝 ( 𝑀 ) ξƒͺ ξƒ­ 1 / π‘ž ≀ 𝐢 β€– π‘Ž β€– βˆ— β€– 𝑓 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) . ( 3 . 1 9 ) Similar to I 1 , we have I 2 ≀ 𝐢 β€– π‘Ž β€– βˆ— β€– 𝑓 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) . ( 3 . 2 0 ) Finally we estimate I I I . The proof of this part is analogue to I , so we just give out an outline. Note that 𝑗 β‰₯ π‘˜ + 2 and π‘₯ ∈ 𝐸 π‘˜ , 𝑦 ∈ 𝐸 𝑗 , | π‘₯ βˆ’ 𝑦 | ∼ | 𝑦 | . So from the condition (3.2), we have β€– β€– ξ€· [ ] 𝑓 π‘Ž , 𝑇 𝑗 ξ€Έ πœ’ π‘˜ β€– β€– 𝐿 𝑝 ( 𝑀 ) ≀ 𝐢 2 βˆ’ 𝑛 𝑗 β€– β€– π‘Ž ( π‘₯ ) βˆ’ π‘Ž 𝐡 π‘˜ , 𝑀 β€– β€– 𝐿 𝑝 ( 𝑀 ) ξ€œ ℝ 𝑛 | | 𝑓 𝑗 ( | | 𝑦 ) 𝑑 𝑦 + 𝐢 2 βˆ’ 𝑛 𝑗 | | | π‘Ž 𝐡 π‘˜ , 𝑀 βˆ’ π‘Ž 𝐡 𝑗 , 𝑀 | | | 𝑀 ξ€· 𝐡 π‘˜ ξ€Έ 1 / 𝑝 ξ€œ ℝ 𝑛 | | 𝑓 𝑗 | | ( 𝑦 ) 𝑑 𝑦 + 𝐢 2 βˆ’ 𝑛 𝑗 𝑀 ξ€· 𝐡 π‘˜ ξ€Έ 1 / 𝑝 ξ€œ ℝ 𝑛 | | | π‘Ž ( 𝑦 ) βˆ’ π‘Ž 𝐡 𝑗 , 𝑀 | | | | | 𝑓 𝑗 | | = 𝐽 ( 𝑦 ) 𝑑 𝑦 ξ…ž 1 + 𝐽 ξ…ž 2 + 𝐽 ξ…ž 3 . ( 3 . 2 1 ) Using hypotheses (iii) for 𝑀 in place of strong doubling, 𝐽 ξ…ž 1 ≀ 𝐢 2 βˆ’ 𝑗 𝑛 β€– π‘Ž β€– βˆ— β€– β€– 𝑓 𝑗 β€– β€– 𝐿 𝑝 ( 𝑀 ) 𝑀 βˆ’ 𝑝 β€² / 𝑝 ξ€· 𝐡 𝑗 ξ€Έ 1 / 𝑝 β€² 𝑀 ξ€· 𝐡 𝑗 ξ€Έ 1 / 𝑝  𝑀 ξ€· 𝐡 π‘˜ ξ€Έ 𝑀 ξ€· 𝐡 𝑗 ξ€Έ ξƒͺ 1 / 𝑝 ≀ 𝐢 2 π‘˜ 𝛿 / 𝑝 2 βˆ’ 𝑗 𝛿 / 𝑝 β€– π‘Ž β€– βˆ— β€– β€– 𝑓 𝑗 β€– β€– 𝐿 𝑝 ( 𝑀 ) . ( 3 . 2 2 ) Similarly, 𝐽 ξ…ž 2 ≀ 𝐢 π‘˜ 2 π‘˜ 𝛿 / 𝑝 𝑗 2 βˆ’ 𝑗 𝛿 / 𝑝 β€– π‘Ž β€– βˆ— β€– β€– 𝑓 𝑗 β€– β€– 𝐿 𝑝 ( 𝑀 ) , 𝐽 ξ…ž 3 ≀ 𝐢 2 π‘˜ 𝛿 / 𝑝 2 βˆ’ 𝑗 𝛿 / 𝑝 β€– π‘Ž β€– βˆ— β€– β€– 𝑓 𝑗 β€– β€– 𝐿 𝑝 ( 𝑀 ) . ( 3 . 2 3 ) Using hypotheses (i) for 𝑀 , that is, πœ‘ ∈ 𝐷 ( 𝑠 , 𝑑 ) , we obtain the following inequality: I I I ≀ 𝐢 β€– π‘Ž β€– βˆ—   π‘˜ 2 π‘˜ ( 𝑠 + 𝛿 / 𝑝 ) π‘ž β‹…   𝑗 β‰₯ π‘˜ + 2 2 𝑗 ( βˆ’ 𝑠 βˆ’ 𝛿 / 𝑝 ) β€– β€– πœ‘ ( 𝑗 ) 𝑓 πœ’ 𝑗 β€– β€– 𝐿 𝑝 ( 𝑀 ) ξƒͺ π‘ž ξƒͺ 1 / π‘ž + 𝐢 β€– π‘Ž β€– βˆ—   π‘˜ ξ€· π‘˜ 2 ( 𝑠 + 𝛿 / 𝑝 ) ξ€Έ π‘ž β‹…   𝑗 β‰₯ π‘˜ + 2 𝑗 2 𝑗 ( βˆ’ 𝑠 βˆ’ 𝛿 / 𝑝 ) β€– β€– πœ‘ ( 𝑗 ) 𝑓 πœ’ 𝑗 β€– β€– 𝐿 𝑝 ( 𝑀 ) ξƒͺ π‘ž ξƒͺ 1 / π‘ž = I I I 1 + I I I 2 . ( 3 . 2 4 ) According to 𝑠 + 𝛿 / 𝑝 > 0 , when π‘ž ≀ 1 , I I I 1 ≀ 𝐢 β€– π‘Ž β€– βˆ—   𝑗 2 𝑗 ( βˆ’ 𝑠 βˆ’ 𝛿 / 𝑝 ) π‘ž πœ‘ ( 𝑗 ) π‘ž β€– β€– 𝑓 πœ’ 𝑗 β€– β€– π‘ž 𝐿 𝑝 ( 𝑀 ) β‹… 𝑗 βˆ’ 2  π‘˜ = βˆ’ ∞ 2 π‘˜ ( 𝑠 + 𝛿 / 𝑝 ) π‘ž ξƒͺ 1 / π‘ž ≀ 𝐢 β€– π‘Ž β€– βˆ— β€– 𝑓 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) . ( 3 . 2 5 ) When π‘ž > 1 , we take πœ€ > 0 such that 𝑠 + 𝛿 / 𝑝 βˆ’ πœ€ > 0 . Then I I I 1 ≀ 𝐢 β€– π‘Ž β€– βˆ— ⎑ ⎒ ⎒ ⎣  π‘˜ 2 π‘˜ ( 𝑠 + 𝛿 / 𝑝 ) π‘ž β‹… βŽ› ⎜ ⎜ ⎝  𝑗 β‰₯ π‘˜ + 2 2 𝑗 ( βˆ’ 𝑠 βˆ’ 𝛿 / 𝑝 + πœ€ ) π‘ž πœ‘ ( 𝑗 ) π‘ž β€– β€– 𝑓 πœ’ 𝑗 β€– β€– π‘ž 𝐿 𝑝 ( 𝑀 ) β‹…   𝑗 β‰₯ π‘˜ + 2 2 βˆ’ 𝑗 π‘ž β€² πœ€ ξƒͺ π‘ž / π‘ž β€² ⎞ ⎟ ⎟ ⎠ ⎀ βŽ₯ βŽ₯ ⎦ 1 / π‘ž ≀ 𝐢 β€– π‘Ž β€– βˆ—   π‘˜ 2 π‘˜ ( 𝑠 + 𝛿 / 𝑝 βˆ’ πœ€ ) π‘ž β‹…   𝑗 β‰₯ π‘˜ + 2 2 𝑗 ( βˆ’ 𝑠 βˆ’ 𝛿 / 𝑝 + πœ€ ) π‘ž πœ‘ ( 𝑗 ) π‘ž β€– β€– 𝑓 πœ’ 𝑗 β€– β€– π‘ž 𝐿 𝑝 ( 𝑀 ) ξƒͺ ξƒ­ 1 / π‘ž ≀ 𝐢 β€– π‘Ž β€– βˆ— β€– 𝑓 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) . ( 3 . 2 6 ) Similar to I I I 1 , we have I I I 2 ≀ 𝐢 β€– π‘Ž β€– βˆ— β€– 𝑓 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) . ( 3 . 2 7 ) This finishes the proof of Theorem 3.2.

The condition (3.2) in Theorem 3.2 can be satisfied by many operators such as Bochner-Riesz operators at the critical index, Ricci-Stein’s oscillatory singular integrals, Fefferman’s multiplier, and the 𝐢 - 𝑍 operators. From this theorem and Theorem 2.7 and  2.10 in [13], we easily deduce the following corollary.

Corollary 3.3. Let 1 < 𝑝 < ∞ , 0 < π‘ž < ∞ , 𝛿 > 0 , and π‘Ž ∈ B M O ( ℝ 𝑛 ) . One assumes that(i) πœ‘ ∈ 𝐷 ( 𝑠 , 𝑑 ) , where βˆ’ ( 𝛿 / 𝑝 ) < 𝑠 ≀ 𝑑 < 𝑛 ( 1 βˆ’ ( 1 / 𝑝 ) ) ,(ii) 𝑀 ∈ 𝐴 π‘Ÿ , where π‘Ÿ = m i n ( 𝑝 , 𝑝 ( 1 βˆ’ ( 𝑑 / 𝑛 ) ) ) ,(iii) 𝑀 ∈ 𝑅 𝐷 ( 𝛿 ) .If 𝐾 is a constant or a variable 𝐢 - 𝑍 kernel on ℝ 𝑛 and 𝑇 is the corresponding 𝐢 - 𝑍 operator, then there exists a constant such that for all 𝑓 ∈ 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( ℝ 𝑛 ) , β€– [ ] 𝑓 β€– π‘Ž , 𝑇 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( ℝ 𝑛 ) ≀ 𝐢 β€– π‘Ž β€– βˆ— β€– 𝑓 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( ℝ 𝑛 ) . ( 3 . 2 8 )

From this and the extension theorem of B M O ( Ξ© ) -functions in [15], by a procedure similar to Theorem  2.11 in [13] and Theorem  2.2 in [16], we can obtain the following corollary.

Corollary 3.4. Let 1 < 𝑝 < ∞ , 0 < π‘ž < ∞ , and 𝛿 > 0 . Suppose that Ξ© is an open set of ℝ 𝑛 and π‘Ž ∈ V M O ( Ξ© ) . One assumes that(i) πœ‘ ∈ 𝐷 ( 𝑠 , 𝑑 ) , where βˆ’ ( 𝛿 / 𝑝 ) < 𝑠 ≀ 𝑑 < 𝑛 ( 1 βˆ’ ( 1 / 𝑝 ) ) ,(ii) 𝑀 ∈ 𝐴 π‘Ÿ , where π‘Ÿ = m i n ( 𝑝 , 𝑝 ( 1 βˆ’ ( 𝑑 / 𝑛 ) ) ) ,(iii) 𝑀 ∈ 𝑅 𝐷 ( 𝛿 ) .If 𝐾 is a variable 𝐢 - 𝑍 kernel on Ξ© and 𝑇 is the corresponding 𝐢 - 𝑍 operator, then for any πœ€ > 0 , there exists a positive number 𝜌 0 = 𝜌 0 ( πœ€ , πœ‚ ) such that for any ball 𝐡 𝑅 with the radius 𝑅 ∈ ( 0 , 𝜌 0 ) , 𝐡 R βŠ† Ξ© and all 𝑓 ∈ 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( 𝐡 𝑅 ) , β€– [ ] 𝑓 β€– π‘Ž , 𝑇 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( 𝐡 𝑅 ) ≀ 𝐢 πœ€ β€– 𝑓 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( 𝐡 𝑅 ) , ( 3 . 2 9 ) where 𝐢 = 𝐢 ( 𝑛 , 𝑝 , π‘ž , π‘Ž , πœ‘ , 𝑀 ) is independent of πœ€ , 𝑓 , and 𝑅 .

4. Interior Estimate of Elliptic Equation

In this section, we will establish the interior regularity of the strong solutions to elliptic equations in weighted Herz spaces by applying the estimates about singular integral operators and linear commutators obtained in the above section.

Suppose that 𝑛 β‰₯ 3 and Ξ© is an open set of ℝ 𝑛 . We are concerned with the nondivergence form elliptic equations β„’ 𝑒 ( π‘₯ ) = βˆ’ 𝑛  𝑖 , 𝑗 = 1 π‘Ž 𝑖 , 𝑗 ( π‘₯ ) 𝑒 π‘₯ 𝑖 π‘₯ 𝑗 = 𝑓 ( π‘₯ ) , a . e . i n Ξ© , ( 4 . 1 ) whose coefficients π‘Ž 𝑖 𝑗 are assumed such that π‘Ž 𝑖 𝑗 ( π‘₯ ) = π‘Ž 𝑗 𝑖 π‘Ž ( π‘₯ ) , a . e . π‘₯ ∈ Ξ© , 𝑖 , 𝑗 = 1 , 2 , … , 𝑛 , 𝑖 𝑗 ∈ 𝐿 ∞ πœ‡ ( Ξ© ) ∩ V M O ( Ξ© ) , βˆ’ 1 | | πœ‰ | | 2 ≀ 𝑛  𝑖 , 𝑗 = 1 π‘Ž 𝑖 , 𝑗 ( π‘₯ ) πœ‰ 𝑖 πœ‰ 𝑗 | | πœ‰ | | ≀ πœ‡ 2 , βˆƒ πœ‡ > 0 , a . e . π‘₯ ∈ Ξ© , πœ‰ ∈ ℝ 𝑛 . ( 4 . 2 ) Let 1 Ξ“ ( π‘₯ , 𝑑 ) = ( 𝑛 βˆ’ 2 ) πœ” 𝑛 ξ€· d e t π‘Ž 𝑖 𝑗 ξ€Έ ( π‘₯ ) 1 / 2  𝑛  𝑖 , 𝑗 = 1 𝐴 𝑖 𝑗 ( π‘₯ ) 𝑑 𝑖 𝑑 𝑗 ξƒͺ ( 2 βˆ’ 𝑛 ) / 2 , Ξ“ 𝑖 πœ• ( π‘₯ , 𝑑 ) = πœ• 𝑑 𝑖 Ξ“ ( π‘₯ , 𝑑 ) , Ξ“ 𝑖 𝑗 πœ• ( π‘₯ , 𝑑 ) = 2 πœ• 𝑑 𝑖 πœ• 𝑑 𝑗 Ξ“ ( π‘₯ , 𝑑 ) , ( 4 . 3 ) for a.e. π‘₯ ∈ 𝐡 and βˆ€ 𝑑 ∈ ℝ 𝑛 ⧡ { 0 } , where the 𝐴 𝑖 𝑗 are the entries of the inverse of the matrix ( π‘Ž 𝑖 𝑗 ) 𝑖 , 𝑗 = 1 , 2 , … , 𝑛 .

From [13], we deduce the interior representation, that is, if 𝑒 ∈ π‘Š 0 2 , 𝑝 , 𝑒 π‘₯ 𝑖 π‘₯ 𝑗 ξ€œ ( π‘₯ ) = P . V . 𝐡 Ξ“ 𝑖 𝑗  ( π‘₯ , π‘₯ βˆ’ 𝑦 ) 𝑛  β„Ž , 𝑙 = 1 ξ€· π‘Ž β„Ž 𝑙 ( π‘₯ ) βˆ’ π‘Ž β„Ž 𝑙 ξ€Έ 𝑒 ( 𝑦 ) π‘₯ β„Ž π‘₯ 𝑙 ξƒ­ ξ€œ ( 𝑦 ) + β„’ 𝑒 ( 𝑦 ) 𝑑 𝑦 + β„’ 𝑒 ( π‘₯ ) | 𝑑 | = 1 Ξ“ 𝑖 ( π‘₯ , 𝑑 ) 𝑑 𝑗 𝑑 𝜎 𝑑 , a . e . f o r π‘₯ ∈ 𝐡 βŠ‚ Ξ© , ( 4 . 4 ) where 𝐡 is a ball in Ξ© . We also set 𝑀 ≑ m a x 𝑖 , 𝑗 = 1 , … , 𝑛 m a x | | 𝛽 | | ≀ 2 𝑛 β€– β€– β€– πœ• 𝛽 πœ• 𝑑 𝛽 Ξ“ 𝑖 𝑗 β€– β€– β€– ( π‘₯ , 𝑑 ) 𝐿 ∞ ( Ξ© Γ— 𝑆 𝑛 βˆ’ 1 ) < ∞ . ( 4 . 5 )

Theorem 4.1. Let 1 < 𝑝 < ∞ , 0 < π‘ž < ∞ , and 𝛿 > 0 . Suppose that Ξ© is an open set of ℝ 𝑛 and π‘Ž 𝑖 𝑗 satisfies (4.2) for 𝑖 , 𝑗 = 1 , 2 , … , 𝑛 . One assumes that(i) πœ‘ ∈ 𝐷 ( 𝑠 , 𝑑 ) , where βˆ’ ( 𝛿 / 𝑝 ) < 𝑠 ≀ 𝑑 < 𝑛 ( 1 βˆ’ ( 1 / 𝑝 ) ) ,(ii) 𝑀 ∈ 𝐴 π‘Ÿ , where π‘Ÿ = m i n ( 𝑝 , 𝑝 ( 1 βˆ’ ( 𝑑 / 𝑛 ) ) ) ,(iii) 𝑀 ∈ 𝑅 𝐷 ( 𝛿 ) .Then there exists a constant 𝐢 such that for all balls 𝐡 βŠ‚ Ξ© and 𝑒 ∈ π‘Š 0 2 , 𝑝 , One has 𝑒 π‘₯ 𝑖 π‘₯ 𝑗 ∈ 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( 𝐡 ) and β€– β€– 𝑒 π‘₯ 𝑖 π‘₯ 𝑗 β€– β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( 𝐡 ) ≀ 𝐢 β€– β„’ 𝑒 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( 𝐡 ) . ( 4 . 6 )

Proof. It is well known that Ξ“ 𝑖 𝑗 ( π‘₯ , 𝑑 ) are 𝐢 - 𝑍 kernels in the 𝑑 variable. Thus, using the technology of [13, 16] and the Corollaries 2.10 and 3.4, we deduce that, for any πœ€ > 0 , β€– β€– 𝑒 π‘₯ 𝑖 π‘₯ 𝑗 β€– β€– 𝐾 q 𝑝 ( πœ‘ , 𝑀 ) ( 𝐡 ) β€– β€– 𝑒 ≀ 𝐢 πœ€ π‘₯ 𝑖 π‘₯ 𝑗 β€– β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( 𝐡 ) + 𝐢 β€– β„’ 𝑒 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( 𝐡 ) . ( 4 . 7 ) Choosing πœ€ to be small enough ( e . g . , 𝐢 πœ€ < 1 ) , we obtain β€– β€– 𝑒 π‘₯ 𝑖 π‘₯ 𝑗 β€– β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( 𝐡 ) ≀ ξ‚΅ 𝐢 ξ‚Ά ( 1 βˆ’ 𝐢 πœ€ ) β€– β„’ 𝑒 β€– 𝐾 π‘ž 𝑝 ( πœ‘ , 𝑀 ) ( 𝐡 ) . ( 4 . 8 ) This finishes the proof of Theorem 4.1.

Acknowledgment

This research is supported by the NSF of China (no. 11161044).

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