# Weighted Herz Spaces and Regularity Results

Authors: Yuxing Guo; Yinsheng Jiang
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

It is proved that, for the nondivergence form elliptic equations∑i,j=1naijuxixj=f, if f belongs to the weighted Herz spaces Kpq(φ,w), then uxixj∈Kpq(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 Kpq(φ,w).


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## Body

## 1. Introduction

For a sequenceφ={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
(1.1)C-12s(k-j)≤φ(k)φ(j)≤C2t(k-j)fork>j.
Let Bk=B(0,2k)={x∈ℝn:|x|≤2k}, Ek=Bk∖Bk-1 for k∈ℤ, and χk=χEk be the characteristic function of the set Ek for k∈ℤ. Suppose that w is a weight function on ℝn. For 1<p<∞, 0<q<∞, the weighted Herz space is defined by
(1.2)Kpq(φ,w)(Rn)={f:fisameasurablefunctionon⁡Rn,‖f‖Kpq(φ,w)<∞},
where
(1.3)‖f‖Kpq(φ,w)=(∑k=-∞∞‍φ(k)q‖fχk‖Lp(w)q)1/q,‖f‖Lp(w)=(∫Rn‍|f(x)|pw(x)dx)1/p.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 Section2, 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 uxixj 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 ∑i,j=1n‍aijuxixj=f, and we prove that if f∈Kpq(φ,w), then uxixj∈Kpq(φ,w), where u is the W2,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.


## 2. Preliminaries

We begin this section with some properties ofAp weights classes which play important role in the proofs of our main results. For more about Ap classes, we can refer to [10, 11].Definition 2.1 (Apweights(1≤p<∞)).
Letw(x)≥0 and w(x)∈Lloc1(ℝn). One says that w∈Ap for 1<p<∞ if there exists a constant C such that for every ball B⊂ℝn,
(2.1)sup⁡B{1|B|∫B‍w(x)dx}{1|B|∫B‍w(x)1-p′dx}p-1≤C
holds, here and below, 1/p+1/p′=1. One says that w∈A1 if there exists a positive constant C such that
(2.2)1|B|∫B‍w(x)dx≤Cessinf⁡x∈B⁡w(x).
The smallest constant appearing in (2.1) or (2.2) is called the Ap constant of w, denoted by Cw.Lemma 2.2.
Let1≤p<∞ and w∈Ap. Then the following statements are true:(1)
(strong doubling) there exists a constantC such that
(2.3)w(Bk)w(Bj)≤C2np(k-j)fork>j,(2)
(centered reverse doubling) for someδ>0, w∈RD(δ), that is,
(2.4)w(Bk)w(Bj)≥C2δ(k-j)fork>j,(3)
for1<p<∞, one has w∈Ap¯   for some p¯<p,(4)
there exist two constantsC and δ>0 such that for any measurable set B⊂E,
(2.5)w(B)w(E)≤C(|B||E|)δ.
If w satisfies (2.5), one says w∈A∞. Obviously, A∞=⋃1≤p<∞‍Ap,(5)
for all(1/p)+(1/p′)=1, one has w1-p′∈Ap′.Remark 2.3.
Note thatw(E)=∫E‍w(x)dx and wp(E)1/p=(∫E‍wp(x)dx)1/p.Definition 2.4.
LetΩ⊂ℝn be an open set. One says that any f∈Lloc1(Ω) is in the bounded mean oscillation spaces BMO(Ω) if
(2.6)sup⁡γ>0,x∈Bγ(x)⊂Ω1|Bγ(x)|∫Bγ(x)‍|f(y)-fBγ(x)|dy≡‖f‖*<∞,
where fBγ(x) is the average over Bγ(x) of f. Moreover, for any f∈BMO(Ω) and r>0, one sets
(2.7)sup⁡γ≤r,x∈Bγ(x)⊂Ω1|Bγ(x)|∫Bγ(x)‍|f(y)-fBγ(x)|dy≡η(r).
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.Remark 2.5.
f
∈
BMO
(
ℝ
n
) or VMO(ℝn) if B ranges in the class of balls of ℝn.Lemma 2.6 (see [12, Theorem 5]).
Letw∈A∞. Then the norm of BMO(w) is equivalent to the norm of BMO(ℝn), where
(2.8)BMO(w)={a:‖a‖*,w=sup⁡1w(B)∫B‍|a(x)-aB,w|w(x)dx},aB,w=1w(B)∫B‍a(z)w(z)dz.Definition 2.7.
LetK:ℝn∖{0}→ℝ. One says that K(x) is a constant Calderón-Zygmund kernel (constant C-Z kernel) if(i)
K∈C∞(ℝn∖{0}),(ii)
K is homogeneous of degree -n,(iii)
∫Sn-1‍K(x)dσ=0, Sn-1={x∈ℝn:|x|=1}.Definition 2.8.
LetΩ be an open set of ℝn and K:Ω×{ℝn∖{0}}→ℝ. One says that K(x,y) is a variable Calderón-Zygmund kernel (variable C-Z kernel) on Ω if(i)
K(x,·) is a constant C-Z kernel for a.e. x∈Ω,(ii)
max|j|≤2n∥(∂j/∂zj)K(x,z)∥L∞(Ω×Sn-1)≡M<∞.
LetK be a constant or a variable C-Z kernel on Ω. One defines the corresponding C-Z operator by
(2.9)Tf(x)=P.V.∫Rn‍K(x-y)f(y)dyorTf(x)=P.V.∫Ω‍K(x,x-y)f(y)dy.Lemma 2.9 (see [5, Theorem 3]).
Let1<p<∞,0<q<∞,δ>0. One assumes that(i)
φ∈D(s,t), where -(δ/p)<s≤t<n(1-(1/p)),(ii)
w∈Ar, where r=min(p,p(1-(t/n))),(iii)
w∈RD(δ).
IfK is a constant or a variable C-Z kernel on ℝn and T is the corresponding C-Z operator, then there exists a constant C such that for all f∈Kpq(φ,w)(ℝn),
(2.10)‖Tf‖Kpq(φ,w)(Rn)≤C‖f‖Kpq(φ,w)(Rn).From this lemma, by a proof similar to that of Theorem  2.11 in [13], we obtain the following corollary.Corollary 2.10.
Let1<p<∞,0<q<∞,δ>0, and Ω be an open set of ℝn. One assumes that(i)
φ∈D(s,t), where -(δ/p)<s≤t<n(1-(1/p)),(ii)
w∈Ar, where r=min(p,p(1-(t/n))),(iii)
w∈RD(δ).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∈Kpq(φ,w)(Ω),
(2.11)‖Tf‖Kpq(φ,w)(Ω)≤C‖f‖Kpq(φ,w)(Ω).


## 3. Weighted Boundedness of Commutators

The aim of this section is to set up the weighted boundedness for the commutators formed byT and BMO(ℝn) 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 ]).
Leta∈BMO(ℝn). Then for any ball B⊂ℝn, there exist constants C1,C2 such that for all α>0,
(3.1)|{x∈B:|a(x)-aB|>α}|≤C1|B|e-C2α/‖a‖*.
The inequality (3.1) is also called John-Nirenberg inequality.Theorem 3.2.
Let1<p<∞,0<q<∞,δ>0, and a∈BMO(ℝn). One assumes that(i)
φ∈D(s,t), where -(δ/p)<s≤t<n(1-(1/p)),(ii)
w∈Ar, where r=min(p,p(1-(t/n))),(iii)
w∈RD(δ).If a linear operator T satisfies
(3.2)|T(f)(x)|≤C∫Rn‍|f(y)||x-y|ndy,x∉supp⁡f,
for any f∈Lloc1(ℝn) and [a,T] is bounded on Lp(w), then [a,T] is also bounded on Kpq(φ,w).Proof.
Letf∈Kpq(φ,w)(ℝn) and a∈BMO(ℝn), we write
(3.3)f(x)=∑j=-∞∞‍f(x)χj(x)=∑j=-∞∞‍fj(x).
Then, we have
(3.4)‖[a,T]f‖Kpq(φ,w)≤C(∑k=-∞∞‍φ(k)q(∑j=-∞k-2‍‖([a,T]fj)χk‖Lp(w))q)1/q+C(∑k=-∞∞‍φ(k)q(∑j=k-1k+1‍‖([a,T]fj)χk‖Lp(w))q)1/q+C(∑k=-∞∞‍φ(k)q(∑j=k+2∞‍‖([a,T]fj)χk‖Lp(w))q)1/q=I+II+III.
For II, by the Lp(w) boundedness of [a,T], we have
(3.5)II≤C(∑k=-∞∞‍φ(k)q(∑j=k-1k+1‍‖a‖*q‖fjχk‖Lp(w))q)1/q≤C‖a‖*‖f‖Kpq(φ,w).
For I, note that when x∈Ek,y∈Ej, and j≤k-2, |x-y|~|x|. So from the condition (3.2), we have
(3.6)|[a,T]fj|≤C∫Rn‍|a(x)-a(y)||x-y|n|fj(y)|dy≤C2-nk|a(x)-aBk,w|∫Rn‍|fj(y)|dy+C2-nk|aBk,w-aBj,w|∫Rn‍|fj(y)|dy+C2-nk∫Rn‍|a(y)-aBj,w‖fj(y)|dy.
Thus,
(3.7)‖([a,T]fj)χk‖Lp(w)≤C2-nk‖(a(x)-aBk,w)χk‖Lp(w)∫Rn‍|fj(y)|dy+C2-nk|aBk,w-aBj,w|w(Bk)1/p∫Rn‍|fj(y)|dy+C2-nkw(Bk)1/p∫Rn‍|a(y)-aBj,w‖fj(y)|dy=J1+J2+J3.
According to Lemma 2.2, w∈Ar¯ for some r¯<r. By Hölder’s inequality and Lemma 2.6,
(3.8)J1≤C2-nk‖a‖*w(Bk)1/p‖fj‖Lp(w)w-p′/p(Bj)1/p′=C2-nk‖a‖*‖fj‖Lp(w)w-p′/p(Bj)1/p′w(Bj)1/p(w(Bk)w(Bj))1/p≤C2-nk‖a‖*‖fj‖Lp(w)|Bj|2nr¯(k-j)/p≤C2k(-n+(r¯n/p))2nj(1-(r¯/p))‖a‖*‖fj‖Lp(w).
It is easy to see that |aBk,w-aBj,w|≤C(k-j)∥a∥*. Therefore, similarly to J1, we have
(3.9)J2≤C(k-j)2-nk‖a‖*w(Bk)1/p‖fj‖Lp(w)w-p′/p(Bj)1/p′≤Ck2k(-n+(r¯n/p))j2nj(1-(r¯/p))‖a‖*‖fj‖Lp(w).
Now, we establish the estimate for term J3,
(3.10)J3≤C2-nkw(Bk)1/p‖fj‖Lp(w)(∫Bj‍|a(y)-aBj,w|p′w1-p′(y)dy)1/p′.
For the simplicity of analysis, we denote H as
(3.11)(∫Bj‍|a(y)-aBj,w|p′w1-p′(y)dy)1/p′.
By an elementary estimate, we have
(3.12)H≤C(∫Bj‍[|a(y)-aBj,w1-p′|+|aBj,w1-p′-aBj,w|]p′w1-p′(y)dy)1/p′≤C‖a‖BMO(w1-p′)w1-p′(Bj)1/p′+|aBj,w1-p′-aBj,w|w1-p′(Bj)1/p′.
Note that
(3.13)|aBj,w1-p′-aBj,w|≤|aBj,w1-p′-aBj|+|aBj-aBj,w|=J31+J32.
Combining (2.5) with (3.1),
(3.14)J32=1w(Bj)∫Bj‍|a(y)-aBj|w(y)dy=1w(Bj)∫0∞‍w({x∈Bj:|a(y)-aBj|>α})dα≤C∫0∞‍e-C2αδ/‖a‖*dα≤C.
In the same manner, we can see that
(3.15)J31≤C.
By Lemma 2.6, we get
(3.16)J3≤C2-nk‖a‖*w(Bk)1/p‖fj‖Lp(w)w-p′/p(Bj)1/p′≤C2k(-n+(r¯n/p))2nj(1-(r¯/p))‖a‖*‖fj‖Lp(w).
Using hypotheses φ∈D(s,t) and the estimates of J1,J2, and J3, we obtain the following inequality:
(3.17)I≤C‖a‖*(∑k‍2k(-n+(r¯n/p)+t)q⋅(∑j≤k-2‍2nj(1-(r¯/p)-(t/n))φ(j)‖fχj‖Lp(w))q)1/q+C‖a‖*(∑k‍(k2k(-n+(r¯n/p)+t))q⋅(∑j≤k-2‍j2nj(1-(r¯/p)-(t/n))φ(j)‖fχj‖Lp(w))q)1/q=I1+I2.
When q≤1, we have
(3.18)I1≤C‖a‖*(∑j‍2nj(1-(r¯/p)-(t/n))qφ(j)q‖fχj‖Lp(w)q⋅∑k=j+2∞‍2k(-n+(r¯n/p)+t)q)1/q≤C‖a‖*‖f‖Kpq(φ,w),
because -n+nr/p+t≤0, that is, -n+nr¯/p+t<0.
Whenq>1, we take ɛ>0 such that -n+nr¯/p+t+nɛ<0. Then
(3.19)I1≤C‖a‖*[∑k‍2k(-n+(r¯n/p)+t)q⋅(∑j=-∞k-2‍2nj(1-(r¯/p)-(t/n)-ɛ)qφ(j)q‖fχj‖Lp(w)q⋅(∑j=-∞k-2‍2nɛq′j)q/q′)]1/q≤C‖a‖*[∑k‍2k(-n+(r¯n/p)+t+nɛ)q⋅(∑j=-∞k-2‍2nj(1-(r¯/p)-(t/n)-ɛ)qφ(j)q‖fχj‖Lp(w)q)]1/q≤C‖a‖*‖f‖Kpq(φ,w).
Similar to I1, we have
(3.20)I2≤C‖a‖*‖f‖Kpq(φ,w).
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∈Ek,y∈Ej, |x-y|~|y|. So from the condition (3.2), we have
(3.21)‖([a,T]fj)χk‖Lp(w)≤C2-nj‖a(x)-aBk,w‖Lp(w)∫Rn‍|fj(y)|dy+C2-nj|aBk,w-aBj,w|w(Bk)1/p∫Rn‍|fj(y)|dy+C2-njw(Bk)1/p∫Rn‍|a(y)-aBj,w||fj(y)|dy=J1′+J2′+J3′.
Using hypotheses (iii) for w in place of strong doubling,
(3.22)J1′≤C2-jn‖a‖*‖fj‖Lp(w)w-p′/p(Bj)1/p′w(Bj)1/p(w(Bk)w(Bj))1/p≤C2kδ/p2-jδ/p‖a‖*‖fj‖Lp(w).
Similarly,
(3.23)J2′≤Ck2kδ/pj2-jδ/p‖a‖*‖fj‖Lp(w),J3′≤C2kδ/p2-jδ/p‖a‖*‖fj‖Lp(w).
Using hypotheses (i) for w, that is, φ∈D(s,t), we obtain the following inequality:
(3.24)III≤C‖a‖*(∑k‍2k(s+δ/p)q⋅(∑j≥k+2‍2j(-s-δ/p)φ(j)‖fχj‖Lp(w))q)1/q+C‖a‖*(∑k‍(k2(s+δ/p))q⋅(∑j≥k+2‍j2j(-s-δ/p)φ(j)‖fχj‖Lp(w))q)1/q=III1+III2.
According to s+δ/p>0, when q≤1,
(3.25)III1≤C‖a‖*(∑j‍2j(-s-δ/p)qφ(j)q‖fχj‖Lp(w)q⋅∑k=-∞j-2‍2k(s+δ/p)q)1/q≤C‖a‖*‖f‖Kpq(φ,w).
When q>1, we take ɛ>0 such that s+δ/p-ɛ>0. Then
(3.26)III1≤C‖a‖*[∑k‍2k(s+δ/p)q⋅(∑j≥k+2‍2j(-s-δ/p+ε)qφ(j)q‖fχj‖Lp(w)q⋅(∑j≥k+2‍2-jq′ε)q/q′)]1/q≤C‖a‖*[∑k‍2k(s+δ/p-ε)q⋅(∑j≥k+2‍2j(-s-δ/p+ε)qφ(j)q‖fχj‖Lp(w)q)]1/q≤C‖a‖*‖f‖Kpq(φ,w).
Similar to III1, we have
(3.27)III2≤C‖a‖*‖f‖Kpq(φ,w).
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.
Let1<p<∞,0<q<∞,δ>0, and a∈BMO(ℝn). One assumes that(i)
φ∈D(s,t), where -(δ/p)<s≤t<n(1-(1/p)),(ii)
w∈Ar, where r=min(p,p(1-(t/n))),(iii)
w∈RD(δ).If K is a constant or a variable C-Z kernel on ℝn and T is the corresponding C-Z operator, then there exists a constant such that for all f∈Kpq(φ,w)(ℝn),
(3.28)‖[a,T]f‖Kpq(φ,w)(Rn)≤C‖a‖*‖f‖Kpq(φ,w)(Rn).From this and the extension theorem ofBMO(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.
Let1<p<∞,0<q<∞, and δ>0. Suppose that Ω is an open set of ℝn and a∈VMO(Ω). One assumes that(i)
φ∈D(s,t), where -(δ/p)<s≤t<n(1-(1/p)),(ii)
w∈Ar, where r=min(p,p(1-(t/n))),(iii)
w∈RD(δ).If K is a variable C-Z kernel on Ω and T is the corresponding C-Z operator, then for any ɛ>0, there exists a positive number ρ0=ρ0(ɛ,η) such that for any ball BR with the radius R∈(0,ρ0), BR⊆Ω and all f∈Kpq(φ,w)(BR),
(3.29)‖[a,T]f‖Kpq(φ,w)(BR)≤Cɛ‖f‖Kpq(φ,w)(BR),
where C=C(n,p,q,a,φ,M) is independent of ɛ,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 thatn≥3 and Ω is an open set of ℝn. We are concerned with the nondivergence form elliptic equations
(4.1)Lu(x)=-∑i,j=1n‍ai,j(x)uxixj=f(x),a.e.inΩ,
whose coefficients aij are assumed such that
(4.2)aij(x)=aji(x),a.e.x∈Ω,i,j=1,2,…,n,aij∈L∞(Ω)∩VMO(Ω),μ-1|ξ|2≤∑i,j=1n‍ai,j(x)ξiξj≤μ|ξ|2,∃μ>0,a.e.x∈Ω,ξ∈Rn.
Let
(4.3)Γ(x,t)=1(n-2)ωn(det⁡aij(x))1/2(∑i,j=1n‍Aij(x)titj)(2-n)/2,Γi(x,t)=∂∂tiΓ(x,t),Γij(x,t)=∂2∂ti∂tjΓ(x,t),
for a.e. x∈B and ∀t∈ℝn∖{0}, where the Aij are the entries of the inverse of the matrix (aij)i,j=1,2,…,n.From [13], we deduce the interior representation, that is, if u∈W02,p,
(4.4)uxixj(x)=P.V.∫B‍Γij(x,x-y)[∑h,l=1n‍(ahl(x)-ahl(y))uxhxl(y)+Lu(y)]dy+Lu(x)∫|t|=1‍Γi(x,t)tjdσt,a.e.forx∈B⊂Ω,
where B is a ball in Ω. We also set
(4.5)M≡max⁡i,j=1,…,nmax⁡|β|≤2n‖∂β∂tβΓij(x,t)‖L∞(Ω×Sn-1)<∞.Theorem 4.1.
Let1<p<∞,0<q<∞, and δ>0. Suppose that Ω is an open set of ℝn and aij 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∈Ar, where r=min(p,p(1-(t/n))),(iii)
w∈RD(δ).Then there exists a constant C such that for all balls B⊂Ω and u∈W02,p, One has uxixj∈Kpq(φ,w)(B) and
(4.6)‖uxixj‖Kpq(φ,w)(B)≤C‖Lu‖Kpq(φ,w)(B).Proof.
It is well known thatΓij(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 ɛ>0,
(4.7)‖uxixj‖Kpq(φ,w)(B)≤Cɛ‖uxixj‖Kpq(φ,w)(B)+C‖Lu‖Kpq(φ,w)(B).
Choosing ɛ to be small enough (e.g.,Cɛ<1), we obtain
(4.8)‖uxixj‖Kpq(φ,w)(B)≤(C(1-Cɛ))‖Lu‖Kpq(φ,w)(B).
This finishes the proof of Theorem 4.1.



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