Weighted Herz Spaces and Regularity Results

It is proved that, for the nondivergence form elliptic equations ∑𝑛𝑖,𝑗=1𝑎𝑖𝑗𝑢𝑥𝑖𝑥𝑗=𝑓, if 𝑓 belongs to the weighted Herz spaces 𝐾𝑞𝑝(𝜑,𝑤), then 𝑢𝑥𝑖𝑥𝑗∈𝐾𝑞𝑝(𝜑,𝑤), where 𝑢 is the 𝑊2,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 𝐾𝑞𝑝(𝜑,𝑤).


Introduction
For a sequence ϕ {ϕ k } ∞ −∞ , ϕ k > 0, we suppose that ϕ satisfies doubling condition of order s, t and write ϕ ∈ D s, t if there exists C ≥ 1 such that and χ k χ E k be the characteristic function of the set E k for k ∈ Z. Suppose that w is a weight function on R n .For 1 < p < ∞, 0 < q < ∞, the weighted Herz space is defined by where

Journal of Function Spaces and Applications
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 Calder ón-Zygmund singular integral operators that will appear in the representation formula of the u x i x 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 n i,j 1 a ij u x i x j f, and we prove that if f ∈ K q p ϕ, w , then u x i x j ∈ K q p ϕ, w , where u is the W 2,p -solution of the equations.Throughout this paper, unless otherwise indicated, C will be used to denote a positive constant that is not necessarily the same at each occurrence.

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 .
holds, here and below, 1/p where f B γ x is the average over B γ x of f.Moreover, for any f ∈ BMO Ω and r > 0, one sets sup One says that any f ∈ BMO Ω is in the vanishing mean oscillation spaces VMO Ω if η r → 0 as r → 0 and refer to η r as the modulus of f.
Let K be a constant or a variable C-Z kernel on Ω.One defines the corresponding C-Z operator by Lemma 2.9 see 5, Theorem 3 .

2.10
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 < q < ∞, δ > 0, and Ω be an open set of R n .One assumes that If K is a constant or a variable C-Z kernel on Ω, and T is the corresponding C-Z operator, then there exists a constant C such that for all f ∈ K q p ϕ, w Ω ,

Weighted Boundedness of Commutators
The aim of this section is to set up the weighted boundedness for the commutators formed by T and BMO R n functions, where a, T f x T af x −a x T f x .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 a ∈ BMO R n .Then for any ball B ⊂ R n , there exist constants C 1 , C 2 such that for all α > 0, 3.1 The inequality 3.1 is also called John-Nirenberg inequality.
Theorem 3.2.Let 1 < p < ∞, 0 < q < ∞, δ > 0, and a ∈ BMO R n .One assumes that If a linear operator T satisfies for any f ∈ L 1 loc R n and a, T is bounded on L p w , then a, T is also bounded on Then, we have I II III.

3.4
For II, by the L p w boundedness of a, T , we have ≤ C a * f K q p ϕ,w .

3.5
For I, note that when x ∈ E k , y ∈ E j , and j ≤ k − 2, |x − y| ∼ |x|.So from the condition 3.2 , we have a y − a B j ,w f j y dy.

3.7
According to Lemma 2.2, w ∈ A r for some r < r.By Hölder's inequality and Lemma 2.6,

3.8
It is easy to see that |a B k ,w − a B j ,w | ≤ C k − j a * .Therefore, similarly to J 1 , we have

3.9
Now, we establish the estimate for term J 3 , a y − a B j ,w p w 1−p y dy 1/p .

3.10
For the simplicity of analysis, we denote H as

3.11
By an elementary estimate, we have

3.13
Combining 2.5 with 3.1 , J 32 1 w B j B j a y − a B j w y dy

3.14
In the same manner, we can see that

3.15
By Lemma 2.6, we get

3.16
Using hypotheses ϕ ∈ D s, t and the estimates of J 1 , J 2 , and J 3 , we obtain the following inequality: 3.17 When q ≤ 1, we have

3.19
Similar to I 1 , we have

3.20
Finally we estimate III.The proof of this part is analogue to I, so we just give out an outline.Note that j ≥ k 2 and x ∈ E k , y ∈ E j , |x − y| ∼ |y|.So from the condition 3.2 , we have

3.21
Using hypotheses iii for w in place of strong doubling,

3.23
Using hypotheses i for w, that is, ϕ ∈ D s, t , we obtain the following inequality:

Journal of Function Spaces and Applications
When q > 1, we take ε > 0 such that s δ/p − ε > 0. Then ≤ C a * f K q p ϕ,w .

3.26
Similar to III 1 , we have III 2 ≤ C a * f K q p ϕ,w .

3.27
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 C-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 < q < ∞, δ > 0, and a ∈ BMO R n .One assumes that i ϕ ∈ D s, t , where − δ/p < s ≤ t < n 1 − 1/p , ii w ∈ A r , where r min p, p 1 − t/n , iii w ∈ RD δ .
If K is a constant or a variable C-Z kernel on R n and T is the corresponding C-Z operator, then there exists a constant such that for all

3.28
From this and the extension theorem of BMO Ω -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 < q < ∞, and δ > 0. Suppose that Ω is an open set of R n and a ∈ VMO Ω .One assumes that If K is a variable C-Z kernel on Ω and T is the corresponding C-Z operator, then for any ε > 0, there exists a positive number ρ 0 ρ 0 ε, η such that for any ball B R with the radius R ∈ 0, ρ where C C n, p, q, a, ϕ, M is independent of ε, f, and R.

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 ≥ 3 and Ω is an open set of R n .We are concerned with the nondivergence form elliptic equations whose coefficients a ij are assumed such that a ij x a ji x , a.e.x ∈ Ω, i, j 1, 2, . . ., n, Then there exists a constant C such that for all balls B ⊂ Ω and u ∈ W 2,p 0 , One has u x i x j ∈ K q p ϕ, w B and u x i x j K q p ϕ,w B ≤ C Lu K q p ϕ,w B .

4.6
Proof.It is well known that Γ ij x, t are C-Z kernels in the t variable.Thus, using the technology of 13, 16 and the Corollaries 2.10 and 3.4, we deduce that, for any ε > 0, u x i x j K q p ϕ,w B ≤ Cε u x i x j K q p ϕ,w B C Lu K q p ϕ,w B .

4.7
Choosing ε to be small enough e.g., Cε < 1 , we obtain This finishes the proof of Theorem 4.1.

Lemma 2.6 see 12
Remark 2.5.f ∈ BMO R n or VMO R n if B ranges in the class of balls of R n .
, Theorem 5 .Let w ∈ A ∞ .Then the norm of BMO w is equivalent to the norm of BMO R n , where BMO w a : a * ,w sup 1 w B B |a x − a B,w |w x dx , ∈ B and ∀t ∈ R n \ {0}, where the A ij are the entries of the inverse of the matrix a ij i,j 1,2,...,n .From 13 , we deduce the interior representation, that is, if u ∈ W Γ i x, t t j dσ t , a.e. for x ∈ B ⊂ Ω, Theorem 4.1.Let 1 < p < ∞, 0 < q < ∞, and δ > 0. Suppose that Ω is an open set of R n and a ij satisfies 4.2 for i, j 1, 2, . . ., n.One assumes that i ϕ ∈ D s, t , where − δ/p < s ≤ t < n 1 − 1/p , ii w ∈ A r , where r min p, p 1 − t/n , iii w ∈ RD δ .